<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
an	O
algorithm	O
is	O
said	O
to	O
be	O
asymptotically	B-General_Concept
optimal	I-General_Concept
if	O
,	O
roughly	O
speaking	O
,	O
for	O
large	O
inputs	O
it	O
performs	O
at	B-General_Concept
worst	I-General_Concept
a	O
constant	O
factor	O
(	O
independent	O
of	O
the	O
input	O
size	O
)	O
worse	O
than	O
the	O
best	O
possible	O
algorithm	O
.	O
</s>
<s>
It	O
is	O
a	O
term	O
commonly	O
encountered	O
in	O
computer	B-General_Concept
science	I-General_Concept
research	O
as	O
a	O
result	O
of	O
widespread	O
use	O
of	O
big-O	O
notation	O
.	O
</s>
<s>
As	O
a	O
simple	O
example	O
,	O
it	O
's	O
known	O
that	O
all	O
comparison	B-Algorithm
sorts	I-Algorithm
require	O
at	O
least	O
comparisons	O
in	O
the	O
average	O
and	O
worst	B-General_Concept
cases	I-General_Concept
.	O
</s>
<s>
Mergesort	B-Algorithm
and	O
heapsort	B-Application
are	O
comparison	B-Algorithm
sorts	I-Algorithm
which	O
perform	O
comparisons	O
,	O
so	O
they	O
are	O
asymptotically	B-General_Concept
optimal	I-General_Concept
in	O
this	O
sense	O
.	O
</s>
<s>
If	O
the	O
input	O
data	O
have	O
some	O
a	O
priori	O
properties	O
which	O
can	O
be	O
exploited	O
in	O
construction	O
of	O
algorithms	O
,	O
in	O
addition	O
to	O
comparisons	O
,	O
then	O
asymptotically	B-General_Concept
faster	I-General_Concept
algorithms	I-General_Concept
may	O
be	O
possible	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
it	O
is	O
known	O
that	O
the	O
objects	O
are	O
integers	O
from	O
the	O
range	O
then	O
they	O
may	O
be	O
sorted	O
time	O
,	O
e.g.	O
,	O
by	O
the	O
bucket	B-Algorithm
sort	I-Algorithm
.	O
</s>
<s>
A	O
consequence	O
of	O
an	O
algorithm	O
being	O
asymptotically	B-General_Concept
optimal	I-General_Concept
is	O
that	O
,	O
for	O
large	O
enough	O
inputs	O
,	O
no	O
algorithm	O
can	O
outperform	O
it	O
by	O
more	O
than	O
a	O
constant	O
factor	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
asymptotically	B-General_Concept
optimal	I-General_Concept
algorithms	I-General_Concept
are	O
often	O
seen	O
as	O
the	O
"	O
end	O
of	O
the	O
line	O
"	O
in	O
research	O
,	O
the	O
attaining	O
of	O
a	O
result	O
that	O
cannot	O
be	O
dramatically	O
improved	O
upon	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
an	O
algorithm	O
is	O
not	O
asymptotically	B-General_Concept
optimal	I-General_Concept
,	O
this	O
implies	O
that	O
as	O
the	O
input	O
grows	O
in	O
size	O
,	O
the	O
algorithm	O
performs	O
increasingly	O
worse	O
than	O
the	O
best	O
possible	O
algorithm	O
.	O
</s>
<s>
Thus	O
asymptotically	B-General_Concept
optimal	I-General_Concept
algorithms	I-General_Concept
are	O
not	O
always	O
the	O
"	O
end	O
of	O
the	O
line	O
"	O
.	O
</s>
<s>
Although	O
asymptotically	B-General_Concept
optimal	I-General_Concept
algorithms	I-General_Concept
are	O
important	O
theoretical	O
results	O
,	O
an	O
asymptotically	B-General_Concept
optimal	I-General_Concept
algorithm	I-General_Concept
might	O
not	O
be	O
used	O
in	O
a	O
number	O
of	O
practical	O
situations	O
:	O
</s>
<s>
It	O
only	O
outperforms	O
more	O
commonly	O
used	O
methods	O
for	O
beyond	O
the	O
range	O
of	O
practical	O
input	O
sizes	O
,	O
such	O
as	O
inputs	O
with	O
more	O
bits	O
than	O
could	O
fit	O
in	O
any	O
computer	B-General_Concept
storage	I-General_Concept
system	I-General_Concept
.	O
</s>
<s>
The	O
inputs	O
encountered	O
in	O
practice	O
fall	O
into	O
special	O
cases	O
that	O
have	O
more	O
efficient	O
algorithms	O
or	O
that	O
heuristic	B-Algorithm
algorithms	I-Algorithm
with	O
bad	O
worst-case	B-General_Concept
times	O
can	O
nevertheless	O
solve	O
efficiently	O
.	O
</s>
<s>
On	O
modern	O
computers	O
,	O
hardware	B-Architecture
optimizations	O
such	O
as	O
memory	B-General_Concept
cache	I-General_Concept
and	O
parallel	B-Operating_System
processing	I-Operating_System
may	O
be	O
"	O
broken	O
"	O
by	O
an	O
asymptotically	B-General_Concept
optimal	I-General_Concept
algorithm	I-General_Concept
(	O
assuming	O
the	O
analysis	O
did	O
not	O
take	O
these	O
hardware	B-Architecture
optimizations	O
into	O
account	O
)	O
.	O
</s>
<s>
An	O
example	O
of	O
an	O
asymptotically	B-General_Concept
optimal	I-General_Concept
algorithm	I-General_Concept
not	O
used	O
in	O
practice	O
is	O
Bernard	O
Chazelle	O
's	O
linear-time	O
algorithm	O
for	O
triangulation	B-Algorithm
of	O
a	O
simple	O
polygon	O
.	O
</s>
<s>
Another	O
is	O
the	O
resizable	B-Data_Structure
array	I-Data_Structure
data	O
structure	O
published	O
in	O
"	O
Resizable	B-Data_Structure
Arrays	I-Data_Structure
in	O
Optimal	O
Time	O
and	O
Space	O
"	O
,	O
which	O
can	O
index	O
in	O
constant	O
time	O
but	O
on	O
many	O
machines	O
carries	O
a	O
heavy	O
practical	O
penalty	O
compared	O
to	O
ordinary	O
array	O
indexing	O
.	O
</s>
<s>
Then	O
,	O
an	O
algorithm	O
which	O
solves	O
the	O
problem	O
in	O
O(f(n )	O
)	O
time	O
is	O
said	O
to	O
be	O
asymptotically	B-General_Concept
optimal	I-General_Concept
.	O
</s>
<s>
Then	O
the	O
algorithm	O
is	O
asymptotically	B-General_Concept
optimal	I-General_Concept
if	O
:	O
</s>
<s>
Although	O
usually	O
applied	O
to	O
time	O
efficiency	O
,	O
an	O
algorithm	O
can	O
be	O
said	O
to	O
use	O
asymptotically	B-General_Concept
optimal	I-General_Concept
space	O
,	O
random	O
bits	O
,	O
number	O
of	O
processors	O
,	O
or	O
any	O
other	O
resource	B-General_Concept
commonly	O
measured	O
using	O
big-O	O
notation	O
.	O
</s>
<s>
Sometimes	O
vague	O
or	O
implicit	O
assumptions	O
can	O
make	O
it	O
unclear	O
whether	O
an	O
algorithm	O
is	O
asymptotically	B-General_Concept
optimal	I-General_Concept
.	O
</s>
<s>
For	O
example	O
,	O
a	O
lower	O
bound	O
theorem	O
might	O
assume	O
a	O
particular	O
abstract	B-Application
machine	I-Application
model	O
,	O
as	O
in	O
the	O
case	O
of	O
comparison	B-Algorithm
sorts	I-Algorithm
,	O
or	O
a	O
particular	O
organization	O
of	O
memory	O
.	O
</s>
<s>
By	O
violating	O
these	O
assumptions	O
,	O
a	O
new	O
algorithm	O
could	O
potentially	O
asymptotically	O
outperform	O
the	O
lower	O
bound	O
and	O
the	O
"	O
asymptotically	B-General_Concept
optimal	I-General_Concept
"	O
algorithms	O
.	O
</s>
<s>
The	O
nonexistence	O
of	O
an	O
asymptotically	B-General_Concept
optimal	I-General_Concept
algorithm	I-General_Concept
is	O
called	O
speedup	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
an	O
open	O
problem	O
whether	O
many	O
of	O
the	O
most	O
well-known	O
algorithms	O
today	O
are	O
asymptotically	B-General_Concept
optimal	I-General_Concept
or	O
not	O
.	O
</s>
<s>
Whether	O
this	O
algorithm	O
is	O
asymptotically	B-General_Concept
optimal	I-General_Concept
is	O
unknown	O
,	O
and	O
would	O
be	O
likely	O
to	O
be	O
hailed	O
as	O
a	O
significant	O
result	O
if	O
it	O
were	O
resolved	O
either	O
way	O
.	O
</s>
