<s>
A	O
zero-suppressed	B-Data_Structure
decision	I-Data_Structure
diagram	I-Data_Structure
(	O
ZSDD	O
or	O
ZDD	B-Data_Structure
)	O
is	O
a	O
particular	O
kind	O
of	O
binary	B-Application
decision	I-Application
diagram	I-Application
(	O
BDD	O
)	O
with	O
fixed	O
variable	O
ordering	O
.	O
</s>
<s>
This	O
data	B-General_Concept
structure	I-General_Concept
provides	O
a	O
canonically	O
compact	O
representation	O
of	O
sets	O
,	O
particularly	O
suitable	O
for	O
certain	O
combinatorial	O
problems	O
.	O
</s>
<s>
Recall	O
the	O
Ordered	B-Application
Binary	I-Application
Decision	I-Application
Diagram	I-Application
(	O
OBDD	B-Application
)	O
reduction	O
strategy	O
,	O
i.e.	O
</s>
<s>
In	O
contrast	O
,	O
a	O
node	O
in	O
a	O
ZDD	B-Data_Structure
is	O
replaced	O
with	O
its	O
negative	O
child	O
if	O
its	O
positive	O
edge	O
points	O
to	O
the	O
terminal	O
node	O
0	O
.	O
</s>
<s>
In	O
a	O
Binary	B-Application
Decision	I-Application
Diagram	I-Application
,	O
a	O
Boolean	O
function	O
can	O
be	O
represented	O
as	O
a	O
rooted	O
,	O
directed	O
,	O
acyclic	O
graph	O
,	O
which	O
consists	O
of	O
several	O
decision	O
nodes	O
and	O
terminal	O
nodes	O
.	O
</s>
<s>
Minato	O
has	O
proved	O
that	O
ZDDs	B-Data_Structure
are	O
especially	O
suitable	O
for	O
combinatorial	O
problems	O
,	O
such	O
as	O
the	O
classical	O
problems	O
in	O
,	O
knight	O
’s	O
tour	O
problem	O
,	O
fault	O
simulation	O
,	O
timing	O
analysis	O
,	O
the	O
N-queens	O
problem	O
,	O
as	O
well	O
as	O
weak	O
division	O
.	O
</s>
<s>
By	O
using	O
ZDDs	B-Data_Structure
,	O
one	O
can	O
reduce	O
the	O
size	O
of	O
the	O
representation	O
of	O
a	O
set	O
of	O
n-bit	O
vectors	O
in	O
OBDDs	B-Application
by	O
at	O
most	O
a	O
factor	O
of	O
n	O
.	O
In	O
practice	O
,	O
the	O
optimization	O
is	O
statistically	B-General_Concept
significant	I-General_Concept
.	O
</s>
<s>
We	O
define	O
a	O
Zero-Suppressed	B-Data_Structure
Decision	I-Data_Structure
Diagram	I-Data_Structure
(	O
ZDD	B-Data_Structure
)	O
to	O
be	O
any	O
directed	O
acyclic	O
graph	O
such	O
that	O
:	O
</s>
<s>
We	O
call	O
Z	O
an	O
unreduced	O
ZDD	B-Data_Structure
,	O
if	O
a	O
HI	O
edge	O
points	O
to	O
a	O
⊥	O
node	O
or	O
condition	O
4	O
fails	O
to	O
hold	O
.	O
</s>
<s>
Let	O
F	O
be	O
a	O
ZDD	B-Data_Structure
.	O
</s>
<s>
Let	O
Fi	O
be	O
the	O
family	O
represented	O
by	O
the	O
ZDD	B-Data_Structure
rooted	O
at	O
vi	O
,	O
which	O
can	O
be	O
shown	O
by	O
proof	O
of	O
induction	O
.	O
</s>
<s>
One	O
feature	O
of	O
ZDDs	B-Data_Structure
is	O
that	O
the	O
form	O
does	O
not	O
depend	O
on	O
the	O
number	O
of	O
input	O
variables	O
as	O
long	O
as	O
the	O
combination	O
sets	O
are	O
the	O
same	O
.	O
</s>
<s>
ZDDs	B-Data_Structure
automatically	O
suppress	O
the	O
variables	O
for	O
objects	O
which	O
never	O
appear	O
in	O
combination	O
,	O
hence	O
the	O
efficiency	O
for	O
manipulating	O
sparse	O
combinations	O
.	O
</s>
<s>
Another	O
advantage	O
of	O
ZDDs	B-Data_Structure
is	O
that	O
the	O
number	O
of	O
1-paths	O
in	O
the	O
graph	O
is	O
exactly	O
equal	O
to	O
the	O
number	O
of	O
elements	O
in	O
the	O
combination	O
set	O
.	O
</s>
<s>
Therefore	O
,	O
ZDDs	B-Data_Structure
are	O
better	O
than	O
simple	O
BDDs	O
to	O
represent	O
combination	O
sets	O
.	O
</s>
<s>
Here	O
we	O
have	O
the	O
basic	O
operations	O
for	O
ZDDs	B-Data_Structure
,	O
as	O
they	O
are	O
slightly	O
different	O
from	O
those	O
of	O
the	O
original	O
BDDs	O
.	O
</s>
<s>
In	O
ZDDs	B-Data_Structure
,	O
there	O
is	O
no	O
NOT	O
operation	O
,	O
which	O
is	O
an	O
essential	O
operation	O
in	O
original	O
BDDs	O
.	O
</s>
<s>
In	O
ZDDs	B-Data_Structure
,	O
can	O
be	O
computed	O
as	O
Diff(U, P )	O
.	O
</s>
<s>
Suppose	O
,	O
we	O
can	O
recursively	O
compute	O
the	O
number	O
of	O
sets	O
in	O
a	O
ZDD	B-Data_Structure
,	O
enabling	O
us	O
to	O
get	O
the	O
34th	O
set	O
out	O
a	O
54-member	O
family	O
.	O
</s>
<s>
Random	O
access	O
is	O
fast	O
,	O
and	O
any	O
operation	O
possible	O
for	O
an	O
array	O
of	O
sets	O
can	O
be	O
done	O
with	O
efficiency	O
on	O
a	O
ZDD	B-Data_Structure
.	O
</s>
<s>
According	O
to	O
Minato	O
,	O
the	O
above	O
operations	O
for	O
ZDDs	B-Data_Structure
can	O
be	O
executed	O
recursively	O
like	O
original	O
BDDs	O
.	O
</s>
<s>
We	O
may	O
use	O
a	O
hash	B-Algorithm
table	I-Algorithm
,	O
called	O
uniq-table	O
,	O
to	O
keep	O
each	O
node	O
unique	O
.	O
</s>
<s>
ZDDs	B-Data_Structure
can	O
be	O
used	O
to	O
represent	O
the	O
five-letter	O
words	O
of	O
English	O
,	O
the	O
set	O
WORDS	O
(	O
of	O
size	O
5757	O
)	O
from	O
the	O
for	O
instance	O
.	O
</s>
<s>
Compared	O
to	O
binary	O
trees	O
,	O
tries	B-General_Concept
,	O
or	O
hash	B-Algorithm
tables	I-Algorithm
,	O
a	O
ZDD	B-Data_Structure
may	O
not	O
be	O
the	O
best	O
to	O
complete	O
simple	O
searches	O
,	O
yet	O
it	O
is	O
efficient	O
in	O
retrieving	O
data	O
that	O
is	O
only	O
partially	O
specified	O
,	O
or	O
data	O
that	O
is	O
only	O
supposed	O
to	O
match	O
a	O
key	O
approximately	O
.	O
</s>
<s>
Moreover	O
,	O
ZDDs	B-Data_Structure
do	O
not	O
involve	O
as	O
many	O
variables	O
.	O
</s>
<s>
In	O
fact	O
,	O
by	O
using	O
a	O
ZDD	B-Data_Structure
,	O
one	O
can	O
represent	O
those	O
five	O
letter	O
words	O
as	O
a	O
sparse	O
function	O
that	O
has	O
26×5	O
=	O
130	O
variables	O
,	O
where	O
variable	O
for	O
example	O
determines	O
whether	O
the	O
second	O
letter	O
is	O
“	O
a	O
”	O
.	O
</s>
<s>
With	O
these	O
130	O
variables	O
the	O
ZDD	B-Data_Structure
size	O
Z(F )	O
is	O
in	O
fact	O
5020	O
instead	O
of	O
6233	O
.	O
</s>
<s>
In	O
spite	O
of	O
having	O
similar	O
theories	O
and	O
algorithms	O
,	O
ZDDs	B-Data_Structure
outperform	O
BDDs	O
for	O
this	O
problem	O
with	O
quite	O
a	O
large	O
margin	O
.	O
</s>
<s>
Consequently	O
,	O
ZDDs	B-Data_Structure
allow	O
us	O
to	O
perform	O
certain	O
queries	O
that	O
are	O
too	O
onerous	O
for	O
BDDs	O
.	O
</s>
<s>
To	O
search	O
words	O
containing	O
a	O
certain	O
pattern	O
,	O
one	O
may	O
use	O
family	O
algebra	O
on	O
ZDDs	B-Data_Structure
to	O
compute	O
where	O
P	O
is	O
the	O
pattern	O
,	O
e.g	O
.	O
</s>
<s>
One	O
may	O
use	O
ZDDs	B-Data_Structure
to	O
represent	O
simple	O
paths	O
in	O
an	O
undirected	O
graph	O
.	O
</s>
<s>
These	O
paths	O
can	O
be	O
represented	O
by	O
the	O
ZDD	B-Data_Structure
shown	O
in	O
Figure	O
13	O
,	O
in	O
which	O
each	O
node	O
mn	O
represents	O
the	O
question	O
"	O
does	O
the	O
path	O
include	O
the	O
arc	O
between	O
m	O
and	O
n	O
?	O
"	O
</s>
<s>
In	O
this	O
ZDD	B-Data_Structure
,	O
we	O
get	O
the	O
first	O
path	O
in	O
Figure	O
12	O
by	O
taking	O
the	O
HI	O
branches	O
at	O
nodes	O
13	O
,	O
36	O
,	O
68	O
,	O
and	O
89	O
of	O
the	O
ZDD	B-Data_Structure
(	O
LO	O
branches	O
that	O
simply	O
go	O
to	O
⊥	O
are	O
omitted	O
)	O
.	O
</s>
<s>
Although	O
the	O
ZDD	B-Data_Structure
in	O
Figure	O
13	O
may	O
not	O
seem	O
significant	O
by	O
any	O
means	O
,	O
the	O
advantages	O
of	O
a	O
ZDD	B-Data_Structure
become	O
obvious	O
as	O
the	O
grid	O
gets	O
larger	O
.	O
</s>
<s>
The	O
paths	O
can	O
be	O
illustrated	O
with	O
33580	O
nodes	O
using	O
a	O
ZDD	B-Data_Structure
.	O
</s>
<s>
One	O
can	O
find	O
a	O
ZDD	B-Data_Structure
that	O
characterizes	O
all	O
paths	O
from	O
CA	O
to	O
ME	O
.	O
</s>
<s>
According	O
to	O
Knuth	O
,	O
this	O
ZDD	B-Data_Structure
turns	O
out	O
to	O
have	O
only	O
7850	O
nodes	O
,	O
and	O
it	O
effectively	O
shows	O
that	O
exactly	O
437,525,772,584	O
simple	O
paths	O
from	O
CA	O
to	O
ME	O
are	O
possible	O
.	O
</s>
<s>
ZDDs	B-Data_Structure
in	O
this	O
case	O
,	O
are	O
efficient	O
for	O
simple	O
paths	O
and	O
Hamiltonian	O
paths	O
.	O
</s>
<s>
Although	O
one	O
can	O
solve	O
this	O
problem	O
by	O
constructing	O
OBDDs	B-Application
,	O
it	O
is	O
more	O
efficient	O
to	O
use	O
ZDDs	B-Data_Structure
.	O
</s>
<s>
Constructing	O
a	O
ZDD	B-Data_Structure
for	O
the	O
8-Queens	O
problem	O
requires	O
8	O
steps	O
from	O
S1	O
to	O
S8	O
.	O
</s>
<s>
The	O
ZDD	B-Data_Structure
for	O
S8	O
consists	O
of	O
all	O
potential	O
solutions	O
of	O
the	O
8-Queens	O
problem	O
.	O
</s>
<s>
According	O
to	O
Minato	O
,	O
the	O
construction	O
of	O
a	O
ZDD	B-Data_Structure
for	O
all	O
solutions	O
is	O
too	O
large	O
to	O
solve	O
directly	O
.	O
</s>
<s>
By	O
dividing	O
the	O
problems	O
into	O
two	O
parts	O
of	O
the	O
board	O
,	O
and	O
constructing	O
ZDDs	B-Data_Structure
in	O
subspaces	O
,	O
one	O
can	O
solve	O
The	O
Knight	O
’s	O
tour	O
problem	O
with	O
each	O
solution	O
containing	O
64	O
edges	O
.	O
</s>
<s>
However	O
,	O
since	O
the	O
graph	O
is	O
not	O
very	O
sparse	O
,	O
the	O
advantage	O
of	O
using	O
ZDDs	B-Data_Structure
is	O
not	O
so	O
obvious	O
.	O
</s>
<s>
N	O
.	O
Takahashi	O
et	O
al	O
suggested	O
a	O
fault	O
simulation	O
method	O
given	O
multiple	O
faults	O
by	O
using	O
OBDDs	B-Application
.	O
</s>
<s>
Since	O
this	O
method	O
involves	O
unate	O
cube	O
set	O
expressions	O
,	O
ZDDs	B-Data_Structure
are	O
more	O
efficient	O
.	O
</s>
<s>
The	O
optimizations	O
from	O
ZDDs	B-Data_Structure
in	O
unate	O
cube	O
set	O
calculations	O
indicate	O
that	O
ZDDs	B-Data_Structure
could	O
be	O
useful	O
in	O
developing	O
VLSI	O
CAD	O
systems	O
and	O
in	O
a	O
myriad	O
of	O
other	O
applications	O
.	O
</s>
<s>
,	O
A	O
CWEB	O
ZDD	B-Data_Structure
implementation	O
by	O
Donald	O
Knuth	O
.	O
</s>
