<s>
In	O
computing	O
,	O
external	B-Application
memory	I-Application
algorithms	I-Application
or	O
out-of-core	B-Application
algorithms	O
are	O
algorithms	O
that	O
are	O
designed	O
to	O
process	O
data	O
that	O
are	O
too	O
large	O
to	O
fit	O
into	O
a	O
computer	O
's	O
main	O
memory	O
at	O
once	O
.	O
</s>
<s>
Such	O
algorithms	O
must	O
be	O
optimized	O
to	O
efficiently	O
fetch	O
and	O
access	O
data	O
stored	O
in	O
slow	O
bulk	O
memory	O
(	O
auxiliary	O
memory	O
)	O
such	O
as	O
hard	B-Device
drives	I-Device
or	O
tape	O
drives	O
,	O
or	O
when	O
memory	O
is	O
on	O
a	O
computer	B-Architecture
network	I-Architecture
.	O
</s>
<s>
External	B-Application
memory	I-Application
algorithms	I-Application
are	O
analyzed	O
in	O
the	O
external	B-Application
memory	I-Application
model	I-Application
.	O
</s>
<s>
External	B-Application
memory	I-Application
algorithms	I-Application
are	O
analyzed	O
in	O
an	O
idealized	O
model	O
of	O
computation	O
called	O
the	O
external	B-Application
memory	I-Application
model	I-Application
(	O
or	O
I/O	B-Application
model	I-Application
,	O
or	O
disk	B-Application
access	I-Application
model	I-Application
)	O
.	O
</s>
<s>
The	O
external	B-Application
memory	I-Application
model	I-Application
is	O
an	O
abstract	B-Application
machine	I-Application
similar	O
to	O
the	O
RAM	B-Application
machine	I-Application
model	I-Application
,	O
but	O
with	O
a	O
cache	B-General_Concept
in	O
addition	O
to	O
main	O
memory	O
.	O
</s>
<s>
The	O
model	O
captures	O
the	O
fact	O
that	O
read	O
and	O
write	O
operations	O
are	O
much	O
faster	O
in	O
a	O
cache	B-General_Concept
than	O
in	O
main	O
memory	O
,	O
and	O
that	O
reading	B-General_Concept
long	O
contiguous	O
blocks	B-General_Concept
is	O
faster	O
than	O
reading	B-General_Concept
randomly	O
using	O
a	O
disk	B-Device
read-and-write	I-Device
head	I-Device
.	O
</s>
<s>
The	O
running	O
time	O
of	O
an	O
algorithm	O
in	O
the	O
external	B-Application
memory	I-Application
model	I-Application
is	O
defined	O
by	O
the	O
number	O
of	O
reads	O
and	O
writes	O
to	O
memory	O
required	O
.	O
</s>
<s>
The	O
external	B-Application
memory	I-Application
model	I-Application
is	O
related	O
to	O
the	O
cache-oblivious	B-Application
model	I-Application
,	O
but	O
algorithms	O
in	O
the	O
external	B-Application
memory	I-Application
model	I-Application
may	O
know	O
both	O
the	O
block	B-General_Concept
size	I-General_Concept
and	O
the	O
cache	B-General_Concept
size	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
the	O
model	O
is	O
sometimes	O
referred	O
to	O
as	O
the	O
cache-aware	O
model	O
.	O
</s>
<s>
The	O
model	O
consists	O
of	O
a	O
processor	O
with	O
an	O
internal	O
memory	O
or	O
cache	B-General_Concept
of	O
size	O
,	O
connected	O
to	O
an	O
unbounded	B-Application
external	O
memory	O
.	O
</s>
<s>
Both	O
the	O
internal	O
and	O
external	O
memory	O
are	O
divided	O
into	O
blocks	B-General_Concept
of	O
size	O
.	O
</s>
<s>
Algorithms	O
in	O
the	O
external	B-Application
memory	I-Application
model	I-Application
take	O
advantage	O
of	O
the	O
fact	O
that	O
retrieving	O
one	O
object	O
from	O
external	O
memory	O
retrieves	O
an	O
entire	O
block	O
of	O
size	O
.	O
</s>
<s>
Searching	O
for	O
an	O
element	O
among	O
objects	O
is	O
possible	O
in	O
the	O
external	B-Application
memory	I-Application
model	I-Application
using	O
a	O
B-tree	B-Architecture
with	O
branching	O
factor	O
.	O
</s>
<s>
Using	O
a	O
B-tree	B-Architecture
,	O
searching	O
,	O
insertion	O
,	O
and	O
deletion	O
can	O
be	O
achieved	O
in	O
time	O
(	O
in	O
Big	O
O	O
notation	O
)	O
.	O
</s>
<s>
Information	O
theoretically	O
,	O
this	O
is	O
the	O
minimum	O
running	O
time	O
possible	O
for	O
these	O
operations	O
,	O
so	O
using	O
a	O
B-tree	B-Architecture
is	O
asymptotically	B-General_Concept
optimal	I-General_Concept
.	O
</s>
<s>
External	B-Algorithm
sorting	I-Algorithm
is	O
sorting	O
in	O
an	O
external	O
memory	O
setting	O
.	O
</s>
<s>
External	B-Algorithm
sorting	I-Algorithm
can	O
be	O
done	O
via	O
distribution	O
sort	O
,	O
which	O
is	O
similar	O
to	O
quicksort	B-Algorithm
,	O
or	O
via	O
a	O
-way	B-Algorithm
merge	I-Algorithm
sort	I-Algorithm
.	O
</s>
<s>
Both	O
variants	O
achieve	O
the	O
asymptotically	B-General_Concept
optimal	I-General_Concept
runtime	O
of	O
to	O
sort	O
objects	O
.	O
</s>
<s>
This	O
bound	O
also	O
applies	O
to	O
the	O
fast	O
Fourier	O
transform	O
in	O
the	O
external	B-Application
memory	I-Application
model	I-Application
.	O
</s>
<s>
The	O
permutation	B-Algorithm
problem	O
is	O
to	O
rearrange	O
elements	O
into	O
a	O
specific	O
permutation	B-Algorithm
.	O
</s>
<s>
Thus	O
,	O
permutation	B-Algorithm
can	O
be	O
done	O
in	O
time	O
.	O
</s>
<s>
The	O
external	B-Application
memory	I-Application
model	I-Application
captures	O
the	O
memory	B-General_Concept
hierarchy	I-General_Concept
,	O
which	O
is	O
not	O
modeled	O
in	O
other	O
common	O
models	O
used	O
in	O
analyzing	O
data	B-General_Concept
structures	I-General_Concept
,	O
such	O
as	O
the	O
random-access	B-Application
machine	I-Application
,	O
and	O
is	O
useful	O
for	O
proving	O
lower	O
bounds	O
for	O
data	B-General_Concept
structures	I-General_Concept
.	O
</s>
<s>
A	O
typical	O
example	O
is	O
geographic	B-Application
information	I-Application
systems	I-Application
,	O
especially	O
digital	B-Application
elevation	I-Application
models	I-Application
,	O
where	O
the	O
full	O
data	O
set	O
easily	O
exceeds	O
several	O
gigabytes	O
or	O
even	O
terabytes	O
of	O
data	O
.	O
</s>
<s>
This	O
methodology	O
extends	O
beyond	O
general	B-General_Concept
purpose	I-General_Concept
CPUs	I-General_Concept
and	O
also	O
includes	O
GPU	B-Architecture
computing	I-Architecture
as	O
well	O
as	O
classical	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
.	O
</s>
<s>
In	O
general-purpose	B-Architecture
computing	I-Architecture
on	I-Architecture
graphics	I-Architecture
processing	I-Architecture
units	I-Architecture
(	O
GPGPU	B-Architecture
)	O
,	O
powerful	O
graphics	O
cards	O
(	O
GPUs	B-Architecture
)	O
with	O
little	O
memory	O
(	O
compared	O
with	O
the	O
more	O
familiar	O
system	O
memory	O
,	O
which	O
is	O
most	O
often	O
referred	O
to	O
simply	O
as	O
RAM	B-Architecture
)	O
are	O
utilized	O
with	O
relatively	O
slow	O
CPU-to-GPU	O
memory	O
transfer	O
(	O
when	O
compared	O
with	O
computation	O
bandwidth	O
)	O
.	O
</s>
<s>
An	O
early	O
use	O
of	O
the	O
term	O
"	O
out-of-core	B-Application
"	O
as	O
an	O
adjective	O
is	O
in	O
1962	O
in	O
reference	O
to	O
devices	O
that	O
are	O
other	O
than	O
the	O
core	O
memory	O
of	O
an	O
IBM	B-Application
360	I-Application
.	O
</s>
<s>
An	O
early	O
use	O
of	O
the	O
term	O
"	O
out-of-core	B-Application
"	O
with	O
respect	O
to	O
algorithms	O
appears	O
in	O
1971	O
.	O
</s>
