Binary Decision Diagram
Overview
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Introduction
Definition of BDD
Why BDD
Reduction Rules
BDD Construction
Optimization of BDD
Types of BDD
Comparison
Applications
Conclusion
Introduction
• Binary decision diagram is a compact data
structure of Boolean logic. It is a graph based
representation of Boolean function. BDD is
defined as directed acyclic graph.
Definition of BDD
A Binary decision diagram is a rooted directed
acyclic graph with
• Two types of vertices-terminal and
non-terminal
• A terminal vertex V has a value(v) € {0,1} and
terminal symbol is □
• A non-terminal vertex V has two children
low(v) and high(V) and non-terminal symbol
is ⃝
Definition of BDD …
f
1
0
0
1
0
0
1
1
Why BDD
• Requires Less Space/Memory. For example,
K-Map requires 2n rows for n input variables
while BDD requires 2n – 1 nodes.
• Efficient ordering algorithm can be applied to
reduce the number of nodes.
Reduction Rules
• Node Elimination
• Node Sharing
Node Elimination
• Eliminating redundant nodes (with both edges
pointing to the same node
a
b
b
Node Sharing
• Merge duplicate nodes (Isomorphic sub
graphs)
f1
f2
a
a
b
c
a
b
c
Construction
F = ac + bc
A
B
C
f
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1
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1
a
b
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c
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c
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c
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c
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Construction …
a
b
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1
Construction …
a
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1
Construction …
a
b
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c
0
1
Construction …
a
b
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c
0
1
Construction …
a
b
c
0
1
Optimization of BDD
Algorithm of grouping input variables
• Grouping of input variables
• Ordering of multiple variables
• Grouping of output function
Optimization of BDD
Three methods exist to reduce the number of
nodes in a BDD
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Derive the dependency matrix
Calculate weight
Construct assignment graph
Obtain the minimum weighted matching
Group the input variables according to minimum
weighted matching
Why Ordering
Bad Ordering can cause significant space problem
ƒ(x1, ..., x8) = x1x2 + x3x4 + x5x6 + x7x8
Why Ordering
Good Ordering of the previous equation
ƒ(x1, ..., x8) = x1x2 + x3x4 + x5x6 + x7x8
Optimization of BDD
Three methods exist to reduce the number of
nodes in a BDD
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Derive the dependency matrix
Calculate weight
Construct assignment graph
Obtain the minimum weighted matching
Group the input variables according to minimum
weighted matching
Dependency Matrix
f0=X1 + X3
f1=X2 X3
f2=X1 X3
f3=X3 + X4
f0
f1
f2
f3
X1
1
0
1
0
X2
0
1
0
0
X3
1
1
1
1
X4
0
0
0
1
Grouping & Weight Calculation
f0
f1
f2
f3
W
[x1 , x2]
1
1
1
0
3
[x1 , x3]
1
1
1
1
4
[x1 , x4]
1
0
1
1
3
[x2 , x3]
1
1
1
1
4
[x2 , x4]
0
1
0
1
2
[x3 , x4]
1
1
1
1
4
Finding Minimum Weight
x1
3
x2
3
2
4
4
x3
4
[x1 , x2] , [x3 , x4]
[x1 , x3] , [x2 , x4]
[x1 , x4] , [x3 , x4]
4+3 = 7
4+2 = 6
4+4 = 8
x4
[x1,x3] , [x2,x4] contains minimum weight 6, hence selected
Types of BDD
Different types of BDD
Shared BDD(SBDD)
Multi-Terminal BDD(MTBDD)
Shared Multi-Terminal BDD
Shared BDD(SBDD)
Representing multiple-output function by
sharing node among BDDs
Reduce the number of nodes
f1=a+b
a
f2=ab
b
0
a
b
1
0
1
Shared BDD(SBDD)
f1=a+b
f2=ab
a
a
b
1
0
Multi-terminal BDD
• Multiple output terminals
• Make simulation faster
a
b
f1
f2
f3
0
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1
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1
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a
b
b
f1
0
1
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f2
1
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f3
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SBDD vs MTBDD
SBDD vs MTBDD
Comparison of Evaluation Time
SBDD vs MTBDD
Comparison of Space Problem
Shared Multi-terminal BDD
[f1, f2]
• Shared internal nodes
• Multiple output terminals
• Group output variables
a
b
f1
f2
f3
f4
0
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1
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[f3, f4]
a
a
b
b
f1
f3
0
0
1
f2
f4
0
1
1
SMTBDD vs SBDD
SMTBDD vs SBDD
Comparison of Evaluation time & Space Problem
SMTBDD vs MTBDD
SMTBDD vs MTBDD
Comparison of Evaluation time & Space Problem
Application of BDD
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Logic Synthesis
Testing
Verification
Fault Tree Analysis
References
• http://en.wikipedia.org/wiki/Binary_decision_diagram
• http://www.columbia.edu/ ~gss2117/
MSRutgersThesis.ppt
• Hafiz Md. Hasan Babu and T.sasao , “Shared Multi
Terminal Binary Decision Diagrams for Multiple-Output
Logic Functions”, the sixth workshop on Synthesis and
System Integration of Mixed Technologies(SASIMI’96),
fukuoka, Japan, pp 212-217 ,Nov 1996.
• S.B Akers, “Binary Decision Diagrams” , IEEE
Trans.Comput; vol C-27, no-6, pp.509-516, June 1978.
Conclusion
BDD are a data structure that has been used for
years to provide a cogent representation of
boolean functions. Indeed, they are now such a
common part of computer aided design
research that one can safely assume a working
knowledge of BDD’s on the part of virtually any
reader