<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
succinct	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
a	O
data	B-General_Concept
structure	I-General_Concept
which	O
uses	O
an	O
amount	O
of	O
space	O
that	O
is	O
"	O
close	O
"	O
to	O
the	O
information-theoretic	O
lower	O
bound	O
,	O
but	O
(	O
unlike	O
other	O
compressed	O
representations	O
)	O
still	O
allows	O
for	O
efficient	O
query	O
operations	O
.	O
</s>
<s>
The	O
concept	O
was	O
originally	O
introduced	O
by	O
Jacobson	O
to	O
encode	O
bit	B-Data_Structure
vectors	I-Data_Structure
,	O
(	O
unlabeled	O
)	O
trees	B-Application
,	O
and	O
planar	O
graphs	O
.	O
</s>
<s>
Unlike	O
general	O
lossless	B-Algorithm
data	I-Algorithm
compression	I-Algorithm
algorithms	O
,	O
succinct	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
retain	O
the	O
ability	O
to	O
use	O
them	O
in-place	O
,	O
without	O
decompressing	O
them	O
first	O
.	O
</s>
<s>
A	O
related	O
notion	O
is	O
that	O
of	O
a	O
compressed	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
,	O
in	O
which	O
the	O
size	O
of	O
the	O
data	B-General_Concept
structure	I-General_Concept
depends	O
upon	O
the	O
particular	O
data	O
being	O
represented	O
.	O
</s>
<s>
implicit	B-Data_Structure
if	O
it	O
takes	O
bits	O
of	O
space	O
,	O
</s>
<s>
For	O
example	O
,	O
a	O
data	B-General_Concept
structure	I-General_Concept
that	O
uses	O
bits	O
of	O
storage	O
is	O
compact	O
,	O
bits	O
is	O
succinct	O
,	O
bits	O
is	O
also	O
succinct	O
,	O
and	O
bits	O
is	O
implicit	B-Data_Structure
.	O
</s>
<s>
Implicit	B-Data_Structure
structures	O
are	O
thus	O
usually	O
reduced	O
to	O
storing	O
information	O
using	O
some	O
permutation	O
of	O
the	O
input	O
data	O
;	O
the	O
most	O
well-known	O
example	O
of	O
this	O
is	O
the	O
heap	B-Application
.	O
</s>
<s>
Succinct	O
indexable	O
dictionaries	O
,	O
also	O
called	O
rank/select	O
dictionaries	O
,	O
form	O
the	O
basis	O
of	O
a	O
number	O
of	O
succinct	O
representation	O
techniques	O
,	O
including	O
binary	O
trees	B-Application
,	O
-ary	O
trees	B-Application
and	O
multisets	B-Language
,	O
as	O
well	O
as	O
suffix	B-Architecture
trees	I-Architecture
and	O
arrays	B-Algorithm
.	O
</s>
<s>
The	O
basic	O
problem	O
is	O
to	O
store	O
a	O
subset	O
of	O
a	O
universe	O
,	O
usually	O
represented	O
as	O
a	O
bit	B-Data_Structure
array	I-Data_Structure
where	O
iff	O
An	O
indexable	O
dictionary	O
supports	O
the	O
usual	O
methods	O
on	O
dictionaries	O
(	O
queries	O
,	O
and	O
insertions/deletions	O
in	O
the	O
dynamic	O
case	O
)	O
as	O
well	O
as	O
the	O
following	O
operations	O
:	O
</s>
<s>
There	O
is	O
a	O
simple	O
representation	O
which	O
uses	O
bits	O
of	O
storage	O
space	O
(	O
the	O
original	O
bit	B-Data_Structure
array	I-Data_Structure
and	O
an	O
auxiliary	O
structure	O
)	O
and	O
supports	O
rank	O
and	O
select	O
in	O
constant	O
time	O
.	O
</s>
<s>
It	O
uses	O
an	O
idea	O
similar	O
to	O
that	O
for	O
range-minimum	B-Algorithm
queries	I-Algorithm
;	O
there	O
are	O
a	O
constant	O
number	O
of	O
recursions	O
before	O
stopping	O
at	O
a	O
subproblem	O
of	O
a	O
limited	O
size	O
.	O
</s>
<s>
The	O
bit	B-Data_Structure
array	I-Data_Structure
is	O
partitioned	O
into	O
large	O
blocks	O
of	O
size	O
bits	O
and	O
small	O
blocks	O
of	O
size	O
bits	O
.	O
</s>
<s>
A	O
lookup	O
table	O
can	O
then	O
be	O
used	O
that	O
stores	O
the	O
answer	O
to	O
every	O
possible	O
rank	O
query	O
on	O
a	O
bit	B-Data_Structure
string	I-Data_Structure
of	O
length	O
for	O
;	O
this	O
requires	O
bits	O
of	O
storage	O
space	O
.	O
</s>
<s>
Thus	O
,	O
since	O
each	O
of	O
these	O
auxiliary	O
tables	O
take	O
space	O
,	O
this	O
data	B-General_Concept
structure	I-General_Concept
supports	O
rank	O
queries	O
in	O
time	O
and	O
bits	O
of	O
space	O
.	O
</s>
<s>
This	O
is	O
often	O
beneficial	O
,	O
since	O
succinct	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
find	O
their	O
uses	O
in	O
large	O
data	O
sets	O
,	O
in	O
which	O
case	O
cache	O
misses	O
become	O
much	O
more	O
frequent	O
and	O
the	O
chances	O
of	O
the	O
lookup	O
table	O
being	O
evicted	O
from	O
closer	O
CPU	O
caches	O
becomes	O
higher	O
.	O
</s>
<s>
A	O
null-terminated	B-Data_Structure
string	I-Data_Structure
(	O
C	B-Language
string	I-Language
)	O
takes	O
Z	O
+	O
1	O
space	O
,	O
and	O
is	O
thus	O
implicit	B-Data_Structure
.	O
</s>
<s>
If	O
there	O
is	O
a	O
maximum	O
length	O
–	O
which	O
is	O
the	O
case	O
in	O
practice	O
,	O
since	O
232	O
=	O
4	O
GiB	O
of	O
data	O
is	O
a	O
very	O
long	O
string	O
,	O
and	O
264	O
=	O
16	O
EiB	O
of	O
data	O
is	O
larger	O
than	O
any	O
string	O
in	O
practice	O
–	O
then	O
a	O
string	O
with	O
a	O
length	O
is	O
also	O
implicit	B-Data_Structure
,	O
taking	O
Z	O
+	O
k	O
space	O
,	O
where	O
k	O
is	O
the	O
number	O
of	O
data	O
to	O
represent	O
the	O
maximum	O
length	O
(	O
e.g.	O
,	O
64	O
bits	O
)	O
.	O
</s>
<s>
An	O
alternative	O
is	O
to	O
place	O
the	O
items	O
in	O
order	O
with	O
a	O
delimiter	O
(	O
e.g.	O
,	O
null-terminated	B-Data_Structure
string	I-Data_Structure
)	O
.	O
</s>
<s>
Another	O
example	O
is	O
the	O
representation	O
of	O
a	O
binary	O
tree	O
:	O
an	O
arbitrary	O
binary	O
tree	O
on	O
nodes	O
can	O
be	O
represented	O
in	O
bits	O
while	O
supporting	O
a	O
variety	O
of	O
operations	O
on	O
any	O
node	O
,	O
which	O
includes	O
finding	O
its	O
parent	O
,	O
its	O
left	O
and	O
right	O
child	O
,	O
and	O
returning	O
the	O
size	O
of	O
its	O
subtree	B-Application
,	O
each	O
in	O
constant	O
time	O
.	O
</s>
<s>
The	O
number	O
of	O
different	O
binary	O
trees	B-Application
on	O
nodes	O
is	O
.	O
</s>
