<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
an	O
enumeration	B-Algorithm
algorithm	I-Algorithm
is	O
an	O
algorithm	O
that	O
enumerates	B-Language
the	O
answers	O
to	O
a	O
computational	O
problem	O
.	O
</s>
<s>
For	O
each	O
input	O
,	O
the	O
enumeration	B-Algorithm
algorithm	I-Algorithm
must	O
produce	O
the	O
list	O
of	O
all	O
solutions	O
,	O
without	O
duplicates	O
,	O
and	O
then	O
halt	O
.	O
</s>
<s>
The	O
performance	O
of	O
an	O
enumeration	B-Algorithm
algorithm	I-Algorithm
is	O
measured	O
in	O
terms	O
of	O
the	O
time	O
required	O
to	O
produce	O
the	O
solutions	O
,	O
either	O
in	O
terms	O
of	O
the	O
total	O
time	O
required	O
to	O
produce	O
all	O
solutions	O
,	O
or	O
in	O
terms	O
of	O
the	O
maximal	O
delay	O
between	O
two	O
consecutive	O
solutions	O
and	O
in	O
terms	O
of	O
a	O
preprocessing	O
time	O
,	O
counted	O
as	O
the	O
time	O
before	O
outputting	O
the	O
first	O
solution	O
.	O
</s>
<s>
This	O
complexity	O
can	O
be	O
expressed	O
in	O
terms	O
of	O
the	O
size	O
of	O
the	O
input	O
,	O
the	O
size	O
of	O
each	O
individual	O
output	O
,	O
or	O
the	O
total	O
size	O
of	O
the	O
set	O
of	O
all	O
outputs	O
,	O
similarly	O
to	O
what	O
is	O
done	O
with	O
output-sensitive	B-Language
algorithms	I-Language
.	O
</s>
<s>
An	O
enumeration	B-Language
problem	O
is	O
defined	O
as	O
a	O
relation	O
over	O
strings	O
of	O
an	O
arbitrary	O
alphabet	O
:	O
</s>
<s>
Enumeration	B-Language
problems	O
have	O
been	O
studied	O
in	O
the	O
context	O
of	O
computational	O
complexity	O
theory	O
,	O
and	O
several	O
complexity	O
classes	O
have	O
been	O
introduced	O
for	O
such	O
problems	O
.	O
</s>
<s>
Polynomial	B-General_Concept
delay	I-General_Concept
,	O
the	O
class	O
of	O
problems	O
where	O
the	O
delay	O
between	O
two	O
consecutive	O
outputs	O
is	O
polynomial	O
in	O
the	O
input	O
(	O
and	O
independent	O
from	O
the	O
output	O
)	O
.	O
</s>
<s>
Strongly	O
polynomial	B-General_Concept
delay	I-General_Concept
,	O
the	O
class	O
of	O
problems	O
where	O
the	O
delay	O
before	O
each	O
output	O
is	O
polynomial	O
in	O
the	O
size	O
of	O
this	O
specific	O
output	O
(	O
and	O
independent	O
from	O
the	O
input	O
or	O
from	O
the	O
other	O
outputs	O
)	O
.	O
</s>
<s>
Backtracking	B-Algorithm
:	O
The	O
simplest	O
way	O
to	O
enumerate	O
all	O
solutions	O
is	O
by	O
systematically	O
exploring	O
the	O
space	O
of	O
possible	O
results	O
(	O
partitioning	O
it	O
at	O
each	O
successive	O
step	O
)	O
.	O
</s>
<s>
However	O
,	O
performing	O
this	O
may	O
not	O
give	O
good	O
guarantees	O
on	O
the	O
delay	O
,	O
i.e.	O
,	O
a	O
backtracking	B-Algorithm
algorithm	I-Algorithm
may	O
spend	O
a	O
long	O
time	O
exploring	O
parts	O
of	O
the	O
space	O
of	O
possible	O
results	O
that	O
do	O
not	O
give	O
rise	O
to	O
a	O
full	O
solution	O
.	O
</s>
<s>
:	O
This	O
technique	O
improves	O
on	O
backtracking	B-Algorithm
by	O
exploring	O
the	O
space	O
of	O
all	O
possible	O
solutions	O
but	O
solving	O
at	O
each	O
step	O
the	O
problem	O
of	O
whether	O
the	O
current	O
partial	O
solution	O
can	O
be	O
extended	O
to	O
a	O
partial	O
solution	O
.	O
</s>
<s>
If	O
the	O
union	O
is	O
non	O
disjoint	O
but	O
the	O
sets	O
can	O
be	O
enumerated	O
in	O
sorted	B-Algorithm
order	I-Algorithm
,	O
then	O
the	O
enumeration	B-Language
can	O
be	O
performed	O
in	O
parallel	O
on	O
both	O
sets	O
while	O
eliminating	O
duplicates	O
on	O
the	O
fly	O
.	O
</s>
<s>
If	O
the	O
union	O
is	O
not	O
disjoint	O
and	O
both	O
sets	O
are	O
not	O
sorted	O
then	O
duplicates	O
can	O
be	O
eliminated	O
at	O
the	O
expense	O
of	O
a	O
higher	O
memory	O
usage	O
,	O
e.g.	O
,	O
using	O
a	O
hash	B-Algorithm
table	I-Algorithm
.	O
</s>
<s>
The	O
vertex	B-Algorithm
enumeration	I-Algorithm
problem	I-Algorithm
,	O
where	O
we	O
are	O
given	O
a	O
polytope	O
described	O
as	O
a	O
system	O
of	O
linear	O
inequalities	O
and	O
we	O
must	O
enumerate	O
the	O
vertices	O
of	O
the	O
polytope	O
.	O
</s>
<s>
This	O
problem	O
is	O
related	O
to	O
monotone	O
dualization	O
and	O
is	O
connected	O
to	O
many	O
applications	O
in	O
database	B-General_Concept
theory	I-General_Concept
and	O
graph	O
theory	O
.	O
</s>
<s>
Enumerating	O
the	O
answers	O
to	O
a	O
database	B-Application
query	I-Application
,	O
for	O
instance	O
a	O
conjunctive	O
query	O
or	O
a	O
query	O
expressed	O
in	O
monadic	O
second-order	O
.	O
</s>
<s>
There	O
have	O
been	O
characterizations	O
in	O
database	B-General_Concept
theory	I-General_Concept
of	O
which	O
conjunctive	O
queries	O
could	O
be	O
enumerated	O
with	O
linear	O
preprocessing	O
and	O
constant	O
delay	O
.	O
</s>
<s>
Several	O
problems	O
on	O
graphs	O
,	O
e.g.	O
,	O
enumerating	O
independent	O
sets	O
,	O
paths	O
,	O
cuts	B-Algorithm
,	O
etc	O
.	O
</s>
<s>
Enumerating	O
the	O
satisfying	O
assignments	O
of	O
representations	O
of	O
Boolean	O
functions	O
,	O
e.g.	O
,	O
a	O
Boolean	O
formula	O
written	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
or	O
disjunctive	B-Application
normal	I-Application
form	I-Application
,	O
a	O
binary	B-Application
decision	I-Application
diagram	I-Application
such	O
as	O
an	O
OBDD	B-Application
,	O
or	O
a	O
Boolean	O
circuit	O
in	O
restricted	O
classes	O
studied	O
in	O
knowledge	B-General_Concept
compilation	I-General_Concept
,	O
e.g.	O
,	O
NNF	B-Application
.	O
</s>
<s>
The	O
notion	O
of	O
enumeration	B-Algorithm
algorithms	I-Algorithm
is	O
also	O
used	O
in	O
the	O
field	O
of	O
computability	O
theory	O
to	O
define	O
some	O
high	O
complexity	O
classes	O
such	O
as	O
RE	O
,	O
the	O
class	O
of	O
all	O
recursively	O
enumerable	O
problems	O
.	O
</s>
<s>
This	O
is	O
the	O
class	O
of	O
sets	O
for	O
which	O
there	O
exist	O
an	O
enumeration	B-Algorithm
algorithm	I-Algorithm
that	O
will	O
produce	O
all	O
elements	O
of	O
the	O
set	O
:	O
the	O
algorithm	O
may	O
run	O
forever	O
if	O
the	O
set	O
is	O
infinite	O
,	O
but	O
each	O
solution	O
must	O
be	O
produced	O
by	O
the	O
algorithm	O
after	O
a	O
finite	O
time	O
.	O
</s>
