<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
binary	B-Application
decision	I-Application
diagram	I-Application
(	O
BDD	O
)	O
or	O
branching	B-Application
program	I-Application
is	O
a	O
data	B-General_Concept
structure	I-General_Concept
that	O
is	O
used	O
to	O
represent	O
a	O
Boolean	O
function	O
.	O
</s>
<s>
On	O
a	O
more	O
abstract	O
level	O
,	O
BDDs	O
can	O
be	O
considered	O
as	O
a	O
compressed	B-General_Concept
representation	O
of	O
sets	O
or	O
relations	O
.	O
</s>
<s>
Unlike	O
other	O
compressed	B-General_Concept
representations	O
,	O
operations	O
are	O
performed	O
directly	O
on	O
the	O
compressed	B-General_Concept
representation	O
,	O
i.e.	O
</s>
<s>
Similar	O
data	B-General_Concept
structures	I-General_Concept
include	O
negation	B-Application
normal	I-Application
form	I-Application
(	O
NNF	O
)	O
,	O
Zhegalkin	O
polynomials	O
,	O
and	O
propositional	O
directed	O
acyclic	O
graphs	O
(	O
PDAG	O
)	O
.	O
</s>
<s>
Each	O
(	O
decision	O
)	O
node	O
is	O
labeled	O
by	O
a	O
Boolean	O
variable	O
and	O
has	O
two	O
child	B-Data_Structure
nodes	I-Data_Structure
called	O
low	O
child	O
and	O
high	O
child	O
.	O
</s>
<s>
In	O
popular	O
usage	O
,	O
the	O
term	O
BDD	O
almost	O
always	O
refers	O
to	O
Reduced	B-Application
Ordered	I-Application
Binary	I-Application
Decision	I-Application
Diagram	I-Application
(	O
ROBDD	B-Application
in	O
the	O
literature	O
,	O
used	O
when	O
the	O
ordering	O
and	O
reduction	O
aspects	O
need	O
to	O
be	O
emphasized	O
)	O
.	O
</s>
<s>
The	O
advantage	O
of	O
an	O
ROBDD	B-Application
is	O
that	O
it	O
is	O
canonical	O
(	O
unique	O
)	O
for	O
a	O
particular	O
function	O
and	O
variable	O
order	O
.	O
</s>
<s>
The	O
left	O
figure	O
below	O
shows	O
a	O
binary	O
decision	B-Algorithm
tree	I-Algorithm
(	O
the	O
reduction	O
rules	O
are	O
not	O
applied	O
)	O
,	O
and	O
a	O
truth	O
table	O
,	O
each	O
representing	O
the	O
function	O
.	O
</s>
<s>
The	O
binary	O
decision	B-Algorithm
tree	I-Algorithm
of	O
the	O
left	O
figure	O
can	O
be	O
transformed	O
into	O
a	O
binary	B-Application
decision	I-Application
diagram	I-Application
by	O
maximally	O
reducing	O
it	O
according	O
to	O
the	O
two	O
reduction	O
rules	O
.	O
</s>
<s>
thumb|546px|Binary	O
decision	B-Algorithm
tree	I-Algorithm
and	O
truth	O
table	O
for	O
the	O
function	O
,	O
described	O
in	O
notation	O
for	O
Boolean	O
operators	O
.	O
</s>
<s>
An	O
ROBDD	B-Application
can	O
be	O
represented	O
even	O
more	O
compactly	O
,	O
using	O
complemented	O
edges	O
.	O
</s>
<s>
The	O
basic	O
idea	O
from	O
which	O
the	O
data	B-General_Concept
structure	I-General_Concept
was	O
created	O
is	O
the	O
Shannon	O
expansion	O
.	O
</s>
<s>
If	O
such	O
a	O
sub-function	O
is	O
considered	O
as	O
a	O
sub-tree	O
,	O
it	O
can	O
be	O
represented	O
by	O
a	O
binary	O
decision	B-Algorithm
tree	I-Algorithm
.	O
</s>
<s>
Binary	B-Application
decision	I-Application
diagrams	I-Application
(	O
BDDs	O
)	O
were	O
introduced	O
by	O
C	O
.	O
Y	O
.	O
Lee	O
,	O
and	O
further	O
studied	O
and	O
made	O
known	O
by	O
Sheldon	O
B	O
.	O
Akers	O
and	O
Raymond	O
T	O
.	O
Boute	O
.	O
</s>
<s>
V	O
.	O
Mamrukov	O
in	O
a	O
CAD	B-Application
for	O
analysis	O
of	O
speed-independent	O
circuits	O
.	O
</s>
<s>
The	O
full	O
potential	O
for	O
efficient	O
algorithms	O
based	O
on	O
the	O
data	B-General_Concept
structure	I-General_Concept
was	O
investigated	O
by	O
Randal	O
Bryant	O
at	O
Carnegie	O
Mellon	O
University	O
:	O
his	O
key	O
extensions	O
were	O
to	O
use	O
a	O
fixed	O
variable	O
ordering	O
(	O
for	O
canonical	O
representation	O
)	O
and	O
shared	O
sub-graphs	O
(	O
for	O
compression	O
)	O
.	O
</s>
<s>
Applying	O
these	O
two	O
concepts	O
results	O
in	O
an	O
efficient	O
data	B-General_Concept
structure	I-General_Concept
and	O
algorithms	O
for	O
the	O
representation	O
of	O
sets	O
and	O
relations	O
.	O
</s>
<s>
one	O
sub-graph	O
is	O
used	O
by	O
several	O
BDDs	O
,	O
the	O
data	B-General_Concept
structure	I-General_Concept
Shared	O
Reduced	B-Application
Ordered	I-Application
Binary	I-Application
Decision	I-Application
Diagram	I-Application
is	O
defined	O
.	O
</s>
<s>
The	O
notion	O
of	O
a	O
BDD	O
is	O
now	O
generally	O
used	O
to	O
refer	O
to	O
that	O
particular	O
data	B-General_Concept
structure	I-General_Concept
.	O
</s>
<s>
In	O
his	O
video	O
lecture	O
Fun	O
With	O
Binary	B-Application
Decision	I-Application
Diagrams	I-Application
(	O
BDDs	O
)	O
,	O
Donald	O
Knuth	O
calls	O
BDDs	O
"	O
one	O
of	O
the	O
only	O
really	O
fundamental	O
data	B-General_Concept
structures	I-General_Concept
that	O
came	O
out	O
in	O
the	O
last	O
twenty-five	O
years	O
"	O
and	O
mentions	O
that	O
Bryant	O
's	O
1986	O
paper	O
was	O
for	O
some	O
time	O
one	O
of	O
the	O
most-cited	O
papers	O
in	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
Another	O
important	O
normal	O
form	O
identified	O
by	O
Darwiche	O
is	O
decomposable	O
negation	B-Application
normal	I-Application
form	I-Application
or	O
DNNF	O
.	O
</s>
<s>
BDDs	O
are	O
extensively	O
used	O
in	O
CAD	B-Application
software	I-Application
to	O
synthesize	O
circuits	O
(	O
logic	O
synthesis	O
)	O
and	O
in	O
formal	O
verification	O
.	O
</s>
<s>
Every	O
arbitrary	O
BDD	O
(	O
even	O
if	O
it	O
is	O
not	O
reduced	O
or	O
ordered	O
)	O
can	O
be	O
directly	O
implemented	O
in	O
hardware	O
by	O
replacing	O
each	O
node	O
with	O
a	O
2	O
to	O
1	O
multiplexer	O
;	O
each	O
multiplexer	O
can	O
be	O
directly	O
implemented	O
by	O
a	O
4-LUT	O
in	O
a	O
FPGA	B-Architecture
.	O
</s>
<s>
It	O
is	O
not	O
so	O
simple	O
to	O
convert	O
from	O
an	O
arbitrary	O
network	O
of	O
logic	O
gates	O
to	O
a	O
BDD	O
(	O
unlike	O
the	O
and-inverter	B-Application
graph	I-Application
)	O
.	O
</s>
<s>
BDDs	O
have	O
been	O
applied	O
in	O
efficient	O
Datalog	B-Language
interpreters	O
.	O
</s>
<s>
It	O
is	O
of	O
crucial	O
importance	O
to	O
care	O
about	O
variable	O
ordering	O
when	O
applying	O
this	O
data	B-General_Concept
structure	I-General_Concept
in	O
practice	O
.	O
</s>
<s>
For	O
any	O
constant	O
c>1	O
it	O
is	O
even	O
NP-hard	O
to	O
compute	O
a	O
variable	O
ordering	O
resulting	O
in	O
an	O
OBDD	B-Application
with	O
a	O
size	O
that	O
is	O
at	O
most	O
c	O
times	O
larger	O
than	O
an	O
optimal	O
one	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
function	O
computing	O
the	O
middle	O
bit	O
of	O
the	O
product	O
of	O
two	O
-bit	O
numbers	O
does	O
not	O
have	O
an	O
OBDD	B-Application
smaller	O
than	O
vertices	O
.	O
</s>
<s>
(	O
If	O
the	O
multiplication	O
function	O
had	O
polynomial-size	O
OBDDs	B-Application
,	O
it	O
would	O
show	O
that	O
integer	O
factorization	O
is	O
in	O
P/poly	O
,	O
which	O
is	O
not	O
known	O
to	O
be	O
true	O
.	O
)	O
</s>
<s>
Researchers	O
have	O
suggested	O
refinements	O
on	O
the	O
BDD	O
data	B-General_Concept
structure	I-General_Concept
giving	O
way	O
to	O
a	O
number	O
of	O
related	O
graphs	O
,	O
such	O
as	O
BMD	O
(	O
binary	B-Application
moment	I-Application
diagrams	I-Application
)	O
,	O
ZDD	B-Data_Structure
(	O
zero-suppressed	B-Data_Structure
decision	I-Data_Structure
diagram	I-Data_Structure
)	O
,	O
FDD	O
(	O
free	O
binary	B-Application
decision	I-Application
diagrams	I-Application
)	O
,	O
PDD	O
(	O
parity	O
decision	O
diagrams	O
)	O
,	O
and	O
MTBDDs	O
(	O
multiple	O
terminal	O
BDDs	O
)	O
.	O
</s>
<s>
Also	O
,	O
since	O
constructing	O
the	O
BDD	O
of	O
a	O
Boolean	O
function	O
solves	O
the	O
NP-complete	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
and	O
the	O
co-NP-complete	O
tautology	O
problem	O
,	O
constructing	O
the	O
BDD	O
can	O
take	O
exponential	O
time	O
in	O
the	O
size	O
of	O
the	O
Boolean	O
formula	O
even	O
when	O
the	O
resulting	O
BDD	O
is	O
small	O
.	O
</s>
