<s>
In	O
the	O
study	O
of	O
graph	O
algorithms	O
,	O
an	O
implicit	B-Data_Structure
graph	I-Data_Structure
representation	I-Data_Structure
(	O
or	O
more	O
simply	O
implicit	B-Data_Structure
graph	I-Data_Structure
)	O
is	O
a	O
graph	O
whose	O
vertices	O
or	O
edges	O
are	O
not	O
represented	O
as	O
explicit	O
objects	O
in	O
a	O
computer	O
's	O
memory	O
,	O
but	O
rather	O
are	O
determined	O
algorithmically	O
from	O
some	O
other	O
input	O
,	O
for	O
example	O
a	O
computable	O
function	O
.	O
</s>
<s>
The	O
notion	O
of	O
an	O
implicit	B-Data_Structure
graph	I-Data_Structure
is	O
common	O
in	O
various	O
search	B-Application
algorithms	I-Application
which	O
are	O
described	O
in	O
terms	O
of	O
graphs	O
.	O
</s>
<s>
In	O
this	O
context	O
,	O
an	O
implicit	B-Data_Structure
graph	I-Data_Structure
may	O
be	O
defined	O
as	O
a	O
set	O
of	O
rules	O
to	O
define	O
all	O
neighbors	O
for	O
any	O
specified	O
vertex	O
.	O
</s>
<s>
This	O
type	O
of	O
implicit	B-Data_Structure
graph	I-Data_Structure
representation	I-Data_Structure
is	O
analogous	O
to	O
an	O
adjacency	B-Data_Structure
list	I-Data_Structure
,	O
in	O
that	O
it	O
provides	O
easy	O
access	O
to	O
the	O
neighbors	O
of	O
each	O
vertex	O
.	O
</s>
<s>
For	O
instance	O
,	O
in	O
searching	O
for	O
a	O
solution	O
to	O
a	O
puzzle	O
such	O
as	O
Rubik	O
's	O
Cube	O
,	O
one	O
may	O
define	O
an	O
implicit	B-Data_Structure
graph	I-Data_Structure
in	O
which	O
each	O
vertex	O
represents	O
one	O
of	O
the	O
possible	O
states	O
of	O
the	O
cube	O
,	O
and	O
each	O
edge	O
represents	O
a	O
move	O
from	O
one	O
state	O
to	O
another	O
.	O
</s>
<s>
In	O
computational	O
complexity	O
theory	O
,	O
several	O
complexity	O
classes	O
have	O
been	O
defined	O
in	O
connection	O
with	O
implicit	B-Data_Structure
graphs	I-Data_Structure
,	O
defined	O
as	O
above	O
by	O
a	O
rule	O
or	O
algorithm	O
for	O
listing	O
the	O
neighbors	O
of	O
a	O
vertex	O
.	O
</s>
<s>
For	O
instance	O
,	O
PPA	O
is	O
the	O
class	O
of	O
problems	O
in	O
which	O
one	O
is	O
given	O
as	O
input	O
an	O
undirected	O
implicit	B-Data_Structure
graph	I-Data_Structure
(	O
in	O
which	O
vertices	O
are	O
-bit	O
binary	O
strings	O
,	O
with	O
a	O
polynomial	O
time	O
algorithm	O
for	O
listing	O
the	O
neighbors	O
of	O
any	O
vertex	O
)	O
and	O
a	O
vertex	O
of	O
odd	O
degree	O
in	O
the	O
graph	O
,	O
and	O
must	O
find	O
a	O
second	O
vertex	O
of	O
odd	O
degree	O
.	O
</s>
<s>
The	O
problem	O
of	O
testing	O
reachability	O
of	O
one	O
vertex	O
to	O
another	O
in	O
an	O
implicit	B-Data_Structure
graph	I-Data_Structure
may	O
also	O
be	O
used	O
to	O
characterize	O
space-bounded	O
nondeterministic	O
complexity	O
classes	O
including	O
NL	O
(	O
the	O
class	O
of	O
problems	O
that	O
may	O
be	O
characterized	O
by	O
reachability	O
in	O
implicit	O
directed	O
graphs	O
whose	O
vertices	O
are	O
-bit	O
bitstrings	O
)	O
,	O
SL	O
(	O
the	O
analogous	O
class	O
for	O
undirected	O
graphs	O
)	O
,	O
and	O
PSPACE	O
(	O
the	O
class	O
of	O
problems	O
that	O
may	O
be	O
characterized	O
by	O
reachability	O
in	O
implicit	B-Data_Structure
graphs	I-Data_Structure
with	O
polynomial-length	O
bitstrings	O
)	O
.	O
</s>
<s>
In	O
this	O
complexity-theoretic	O
context	O
,	O
the	O
vertices	O
of	O
an	O
implicit	B-Data_Structure
graph	I-Data_Structure
may	O
represent	O
the	O
states	O
of	O
a	O
nondeterministic	O
Turing	O
machine	O
,	O
and	O
the	O
edges	O
may	O
represent	O
possible	O
state	O
transitions	O
,	O
but	O
implicit	B-Data_Structure
graphs	I-Data_Structure
may	O
also	O
be	O
used	O
to	O
represent	O
many	O
other	O
types	O
of	O
combinatorial	O
structure	O
.	O
</s>
<s>
PLS	O
,	O
another	O
complexity	O
class	O
,	O
captures	O
the	O
complexity	O
of	O
finding	O
local	O
optima	O
in	O
an	O
implicit	B-Data_Structure
graph	I-Data_Structure
.	O
</s>
<s>
Implicit	B-Data_Structure
graph	I-Data_Structure
models	O
have	O
also	O
been	O
used	O
as	O
a	O
form	O
of	O
relativization	O
in	O
order	O
to	O
prove	O
separations	O
between	O
complexity	O
classes	O
that	O
are	O
stronger	O
than	O
the	O
known	O
separations	O
for	O
non-relativized	O
models	O
.	O
</s>
<s>
used	O
neighborhood	O
representations	O
of	O
implicit	B-Data_Structure
graphs	I-Data_Structure
to	O
define	O
a	O
graph	O
traversal	O
problem	O
that	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
on	O
a	O
quantum	B-Architecture
computer	I-Architecture
but	O
that	O
requires	O
exponential	O
time	O
to	O
solve	O
on	O
any	O
classical	O
computer	O
.	O
</s>
<s>
That	O
is	O
,	O
this	O
type	O
of	O
implicit	O
representation	O
is	O
analogous	O
to	O
an	O
adjacency	B-Algorithm
matrix	I-Algorithm
:	O
it	O
is	O
straightforward	O
to	O
check	O
whether	O
two	O
vertices	O
are	O
adjacent	O
but	O
finding	O
the	O
neighbors	O
of	O
any	O
vertex	O
may	O
involve	O
looping	O
through	O
all	O
vertices	O
and	O
testing	O
which	O
ones	O
are	O
neighbors	O
.	O
</s>
<s>
More	O
generally	O
,	O
the	O
same	O
approach	O
can	O
be	O
used	O
to	O
provide	O
an	O
implicit	O
representation	O
for	O
graphs	O
with	O
bounded	O
arboricity	O
or	O
bounded	O
degeneracy	B-Algorithm
,	O
including	O
the	O
planar	O
graphs	O
and	O
the	O
graphs	O
in	O
any	O
minor-closed	O
graph	O
family	O
.	O
</s>
<s>
For	O
this	O
application	O
of	O
implicit	B-Data_Structure
graph	I-Data_Structure
representations	I-Data_Structure
,	O
it	O
is	O
important	O
that	O
the	O
labels	O
use	O
as	O
few	O
bits	O
as	O
possible	O
,	O
because	O
the	O
number	O
of	O
bits	O
in	O
the	O
labels	O
translates	O
directly	O
into	O
the	O
number	O
of	O
vertices	O
in	O
the	O
induced	O
universal	O
graph	O
.	O
</s>
<s>
The	O
Aanderaa	O
–	O
Karp	O
–	O
Rosenberg	O
conjecture	O
concerns	O
implicit	B-Data_Structure
graphs	I-Data_Structure
given	O
as	O
a	O
set	O
of	O
labeled	O
vertices	O
with	O
a	O
black-box	O
rule	O
for	O
determining	O
whether	O
any	O
two	O
vertices	O
are	O
adjacent	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
rule	O
could	O
be	O
to	O
look	O
up	O
the	O
pair	O
of	O
vertices	O
in	O
a	O
separate	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
However	O
,	O
an	O
algorithm	O
that	O
is	O
given	O
as	O
input	O
an	O
implicit	B-Data_Structure
graph	I-Data_Structure
of	O
this	O
type	O
must	O
operate	O
on	O
it	O
only	O
through	O
the	O
implicit	O
adjacency	O
test	O
,	O
without	O
reference	O
to	O
how	O
the	O
test	O
is	O
implemented	O
.	O
</s>
<s>
In	O
this	O
context	O
,	O
the	O
question	O
to	O
be	O
determined	O
is	O
how	O
many	O
pairs	O
of	O
vertices	O
must	O
be	O
tested	O
for	O
adjacency	O
,	O
in	O
the	O
worst	O
case	O
,	O
before	O
the	O
property	O
of	O
interest	O
can	O
be	O
determined	O
to	O
be	O
true	O
or	O
false	O
for	O
a	O
given	O
implicit	B-Data_Structure
graph	I-Data_Structure
.	O
</s>
<s>
In	O
contrast	O
,	O
such	O
a	O
fast	O
time	O
is	O
not	O
possible	O
in	O
neighborhood-based	O
implicit	B-Data_Structure
graph	I-Data_Structure
models	O
,	O
</s>
