<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
an	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
or	O
space-efficient	O
data	B-General_Concept
structure	I-General_Concept
is	O
a	O
data	B-General_Concept
structure	I-General_Concept
that	O
stores	O
very	O
little	O
information	O
other	O
than	O
the	O
main	O
or	O
required	O
data	O
:	O
a	O
data	B-General_Concept
structure	I-General_Concept
that	O
requires	O
low	O
overhead	O
.	O
</s>
<s>
A	O
less	O
restrictive	O
definition	O
is	O
a	O
succinct	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
,	O
which	O
allows	O
greater	O
overhead	O
.	O
</s>
<s>
An	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
one	O
with	O
constant	O
space	O
overhead	O
(	O
above	O
the	O
information-theoretic	O
lower	O
bound	O
)	O
.	O
</s>
<s>
Historically	O
,	O
defined	O
an	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
(	O
and	O
algorithms	O
acting	O
on	O
one	O
)	O
as	O
one	O
"	O
in	O
which	O
structural	O
information	O
is	O
implicit	O
in	O
the	O
way	O
data	O
are	O
stored	O
,	O
rather	O
than	O
explicit	O
in	O
pointers.	O
"	O
</s>
<s>
They	O
are	O
somewhat	O
vague	O
in	O
the	O
definition	O
,	O
defining	O
it	O
most	O
strictly	O
as	O
a	O
single	O
array	O
,	O
with	O
only	O
the	O
size	O
retained	O
(	O
a	O
single	O
number	O
of	O
overhead	O
)	O
,	O
or	O
more	O
loosely	O
as	O
a	O
data	B-General_Concept
structure	I-General_Concept
with	O
constant	O
overhead	O
(	O
)	O
.	O
</s>
<s>
This	O
latter	O
definition	O
is	O
today	O
more	O
standard	O
,	O
and	O
the	O
still-looser	O
notion	O
of	O
a	O
data	B-General_Concept
structure	I-General_Concept
with	O
non-constant	O
but	O
small	O
overhead	O
is	O
today	O
known	O
as	O
a	O
succinct	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
,	O
as	O
defined	O
by	O
;	O
it	O
was	O
referred	O
to	O
as	O
semi-implicit	O
by	O
.	O
</s>
<s>
A	O
fundamental	O
distinction	O
is	O
between	O
static	O
data	B-General_Concept
structures	I-General_Concept
(	O
read-only	O
)	O
and	O
dynamic	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
(	O
which	O
can	O
be	O
modified	O
)	O
.	O
</s>
<s>
Simple	O
implicit	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
,	O
such	O
as	O
representing	O
a	O
sorted	O
list	O
as	O
an	O
array	O
,	O
may	O
be	O
very	O
efficient	O
as	O
a	O
static	O
data	B-General_Concept
structure	I-General_Concept
,	O
but	O
inefficient	O
as	O
a	O
dynamic	O
data	B-General_Concept
structure	I-General_Concept
,	O
due	O
to	O
modification	O
operations	O
(	O
such	O
as	O
insertion	O
in	O
the	O
case	O
of	O
a	O
sorted	O
list	O
)	O
being	O
inefficient	O
.	O
</s>
<s>
A	O
trivial	O
example	O
of	O
an	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
an	O
array	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
,	O
which	O
is	O
an	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
for	O
a	O
list	O
,	O
and	O
requires	O
only	O
the	O
constant	O
overhead	O
of	O
the	O
length	O
;	O
unlike	O
a	O
linked	B-Data_Structure
list	I-Data_Structure
,	O
which	O
has	O
a	O
pointer	O
associated	O
with	O
each	O
data	O
element	O
,	O
which	O
explicitly	O
gives	O
the	O
relationship	O
from	O
one	O
element	O
to	O
the	O
next	O
.	O
</s>
<s>
Similarly	O
,	O
a	O
null-terminated	B-Data_Structure
string	I-Data_Structure
is	O
an	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
for	O
a	O
string	O
(	O
list	O
of	O
characters	O
)	O
.	O
</s>
<s>
These	O
are	O
considered	O
very	O
simple	O
because	O
they	O
are	O
static	O
data	B-General_Concept
structures	I-General_Concept
(	O
read-only	O
)	O
,	O
and	O
only	O
admit	O
the	O
simple	O
operation	O
of	O
iteration	O
over	O
the	O
elements	O
.	O
</s>
<s>
The	O
elements	O
need	O
not	O
be	O
of	O
the	O
same	O
type	O
,	O
and	O
a	O
table	B-Application
of	O
data	O
(	O
a	O
list	O
of	O
records	O
)	O
may	O
similarly	O
be	O
represented	O
implicitly	O
as	O
a	O
flat	O
(	O
1-dimensional	O
)	O
list	O
,	O
together	O
with	O
the	O
length	O
of	O
each	O
field	B-Application
,	O
so	O
long	O
as	O
each	O
field	B-Application
has	O
uniform	O
size	O
(	O
so	O
a	O
single	O
size	O
can	O
be	O
used	O
per	O
field	B-Application
,	O
not	O
per	O
record	O
)	O
.	O
</s>
<s>
A	O
less	O
trivial	O
example	O
is	O
representing	O
a	O
sorted	O
list	O
by	O
a	O
sorted	B-Data_Structure
array	I-Data_Structure
,	O
which	O
allows	O
search	O
in	O
logarithmic	O
time	O
by	O
binary	O
search	O
.	O
</s>
<s>
Contrast	O
with	O
a	O
search	B-Data_Structure
tree	I-Data_Structure
,	O
specifically	O
a	O
binary	B-Language
search	I-Language
tree	I-Language
,	O
which	O
also	O
allows	O
logarithmic-time	O
search	O
,	O
but	O
requires	O
pointers	O
.	O
</s>
<s>
A	O
sorted	B-Data_Structure
array	I-Data_Structure
is	O
only	O
efficient	O
as	O
a	O
static	O
data	B-General_Concept
structure	I-General_Concept
,	O
as	O
modifying	O
the	O
list	O
is	O
slow	O
–	O
unlike	O
a	O
binary	B-Language
search	I-Language
tree	I-Language
–	O
but	O
does	O
not	O
require	O
the	O
space	O
overhead	O
of	O
a	O
tree	O
.	O
</s>
<s>
An	O
important	O
example	O
of	O
an	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
representing	O
a	O
perfect	O
binary	O
tree	O
as	O
a	O
list	O
,	O
in	O
increasing	O
order	O
of	O
depth	O
,	O
so	O
root	O
,	O
first	O
left	O
child	O
,	O
first	O
right	O
child	O
,	O
first	O
left	O
child	O
of	O
first	O
left	O
child	O
,	O
etc	O
.	O
</s>
<s>
Such	O
a	O
tree	O
occurs	O
notably	O
for	O
an	O
ancestry	O
chart	O
to	O
a	O
given	O
depth	O
,	O
and	O
the	O
implicit	O
representation	O
is	O
known	O
as	O
an	O
Ahnentafel	O
(	O
ancestor	O
table	B-Application
)	O
.	O
</s>
<s>
This	O
can	O
be	O
generalized	O
to	O
a	O
complete	O
binary	O
tree	O
(	O
where	O
the	O
last	O
level	O
may	O
be	O
incomplete	O
)	O
,	O
which	O
yields	O
the	O
best-known	O
example	O
of	O
an	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
,	O
namely	O
the	O
binary	B-Application
heap	I-Application
,	O
which	O
is	O
an	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
for	O
a	O
priority	B-Application
queue	I-Application
.	O
</s>
<s>
This	O
is	O
more	O
sophisticated	O
than	O
earlier	O
examples	O
because	O
it	O
allows	O
multiple	O
operations	O
,	O
and	O
is	O
an	O
efficient	O
dynamic	O
data	B-General_Concept
structure	I-General_Concept
(	O
it	O
allows	O
efficient	O
modification	O
of	O
the	O
data	O
)	O
:	O
not	O
only	O
top	O
,	O
but	O
also	O
insert	O
and	O
pop	O
.	O
</s>
<s>
More	O
sophisticated	O
implicit	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
include	O
the	O
beap	B-Application
(	O
bi-parental	B-Application
heap	I-Application
)	O
.	O
</s>
<s>
The	O
trivial	O
examples	O
of	O
lists	O
or	O
tables	O
of	O
values	O
date	O
to	O
prehistory	O
,	O
while	O
historically	O
non-trivial	O
implicit	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
date	O
at	O
least	O
to	O
the	O
Ahnentafel	O
,	O
which	O
was	O
introduced	O
by	O
Michaël	O
Eytzinger	O
in	O
1590	O
for	O
use	O
in	O
genealogy	O
.	O
</s>
<s>
In	O
formal	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
first	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
generally	O
considered	O
to	O
be	O
the	O
sorted	O
list	O
,	O
used	O
for	O
binary	O
search	O
,	O
which	O
was	O
introduced	O
by	O
John	O
Mauchly	O
in	O
1946	O
,	O
in	O
the	O
Moore	O
School	O
Lectures	O
,	O
the	O
first	O
ever	O
set	O
of	O
lectures	O
regarding	O
any	O
computer-related	O
topic	O
.	O
</s>
<s>
The	O
binary	B-Application
heap	I-Application
was	O
introduced	O
in	O
to	O
implement	O
the	O
heapsort	B-Application
.	O
</s>
<s>
The	O
notion	O
of	O
an	O
implicit	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
was	O
formalized	O
in	O
,	O
as	O
part	O
of	O
introducing	O
and	O
analyzing	O
the	O
beap	B-Application
.	O
</s>
