<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
integer	B-Algorithm
sorting	I-Algorithm
is	O
the	O
algorithmic	O
problem	O
of	O
sorting	B-Algorithm
a	O
collection	B-Application
of	O
data	O
values	O
by	O
integer	O
keys	O
.	O
</s>
<s>
Algorithms	O
designed	O
for	O
integer	B-Algorithm
sorting	I-Algorithm
may	O
also	O
often	O
be	O
applied	O
to	O
sorting	B-Algorithm
problems	I-Algorithm
in	O
which	O
the	O
keys	O
are	O
floating	B-Algorithm
point	I-Algorithm
numbers	I-Algorithm
,	O
rational	O
numbers	O
,	O
or	O
text	O
strings	O
.	O
</s>
<s>
The	O
ability	O
to	O
perform	O
integer	O
arithmetic	O
on	O
the	O
keys	O
allows	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
to	O
be	O
faster	O
than	O
comparison	B-Algorithm
sorting	I-Algorithm
algorithms	O
in	O
many	O
cases	O
,	O
depending	O
on	O
the	O
details	O
of	O
which	O
operations	O
are	O
allowed	O
in	O
the	O
model	O
of	O
computing	O
and	O
how	O
large	O
the	O
integers	O
to	O
be	O
sorted	O
are	O
.	O
</s>
<s>
Integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
including	O
pigeonhole	B-Algorithm
sort	I-Algorithm
,	O
counting	B-Algorithm
sort	I-Algorithm
,	O
and	O
radix	B-Algorithm
sort	I-Algorithm
are	O
widely	O
used	O
and	O
practical	O
.	O
</s>
<s>
Other	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
with	O
smaller	O
worst-case	B-General_Concept
time	O
bounds	O
are	O
not	O
believed	O
to	O
be	O
practical	O
for	O
computer	O
architectures	O
with	O
64	O
or	O
fewer	O
bits	O
per	O
word	O
.	O
</s>
<s>
Many	O
such	O
algorithms	O
are	O
known	O
,	O
with	O
performance	O
depending	O
on	O
a	O
combination	O
of	O
the	O
number	O
of	O
items	O
to	O
be	O
sorted	O
,	O
number	O
of	O
bits	O
per	O
key	O
,	O
and	O
number	O
of	O
bits	O
per	O
word	O
of	O
the	O
computer	O
performing	O
the	O
sorting	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Time	O
bounds	O
for	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
typically	O
depend	O
on	O
three	O
parameters	O
:	O
the	O
number	O
of	O
data	O
values	O
to	O
be	O
sorted	O
,	O
the	O
magnitude	O
of	O
the	O
largest	O
possible	O
key	O
to	O
be	O
sorted	O
,	O
and	O
the	O
number	O
of	O
bits	O
that	O
can	O
be	O
represented	O
in	O
a	O
single	O
machine	O
word	O
of	O
the	O
computer	O
on	O
which	O
the	O
algorithm	O
is	O
to	O
be	O
performed	O
.	O
</s>
<s>
Integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
are	O
usually	O
designed	O
to	O
work	O
in	O
either	O
the	O
pointer	B-Application
machine	I-Application
or	O
random	B-Application
access	I-Application
machine	I-Application
models	O
of	O
computing	O
.	O
</s>
<s>
The	O
random	B-Application
access	I-Application
machine	I-Application
allows	O
any	O
value	O
that	O
is	O
stored	O
in	O
a	O
register	O
to	O
be	O
used	O
as	O
the	O
address	O
of	O
memory	O
read	O
and	O
write	O
operations	O
,	O
with	O
unit	O
cost	O
per	O
operation	O
.	O
</s>
<s>
In	O
contrast	O
,	O
in	O
the	O
pointer	B-Application
machine	I-Application
model	O
,	O
read	O
and	O
write	O
operations	O
use	O
addresses	O
stored	O
in	O
pointers	O
,	O
and	O
it	O
is	O
not	O
allowed	O
to	O
perform	O
arithmetic	O
operations	O
on	O
these	O
pointers	O
.	O
</s>
<s>
Different	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
make	O
different	O
assumptions	O
,	O
however	O
,	O
about	O
whether	O
integer	O
multiplication	O
is	O
also	O
allowed	O
as	O
a	O
unit-time	O
operation	O
.	O
</s>
<s>
Other	O
more	O
specialized	O
models	O
of	O
computation	O
such	O
as	O
the	O
parallel	B-Operating_System
random	I-Operating_System
access	I-Operating_System
machine	I-Operating_System
have	O
also	O
been	O
considered	O
.	O
</s>
<s>
showed	O
that	O
in	O
some	O
cases	O
the	O
multiplications	O
or	O
table	O
lookups	O
required	O
by	O
some	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
could	O
be	O
replaced	O
by	O
customized	O
operations	O
that	O
would	O
be	O
more	O
easily	O
implemented	O
in	O
hardware	O
but	O
that	O
are	O
not	O
typically	O
available	O
on	O
general-purpose	O
computers	O
.	O
</s>
<s>
improved	O
on	O
this	O
by	O
showing	O
how	O
to	O
replace	O
these	O
special	O
operations	O
by	O
the	O
bit	O
field	O
manipulation	O
instructions	O
already	O
available	O
on	O
Pentium	B-General_Concept
processors	O
.	O
</s>
<s>
In	O
external	B-Application
memory	I-Application
models	I-Application
of	I-Application
computing	I-Application
,	O
no	O
known	O
integer	B-Algorithm
sorting	I-Algorithm
algorithm	O
is	O
faster	O
than	O
comparison	B-Algorithm
sorting	I-Algorithm
.	O
</s>
<s>
Researchers	O
have	O
shown	O
that	O
,	O
in	O
these	O
models	O
,	O
restricted	O
classes	O
of	O
algorithms	O
that	O
are	O
limited	O
in	O
how	O
they	O
manipulate	O
their	O
keys	O
cannot	O
be	O
faster	O
than	O
comparison	B-Algorithm
sorting	I-Algorithm
,	O
and	O
that	O
an	O
integer	B-Algorithm
sorting	I-Algorithm
algorithm	O
that	O
is	O
faster	O
than	O
comparison	B-Algorithm
sorting	I-Algorithm
would	O
imply	O
the	O
falsity	O
of	O
a	O
standard	O
conjecture	O
in	O
network	O
coding	O
.	O
</s>
<s>
A	O
priority	B-Application
queue	I-Application
is	O
a	O
data	O
structure	O
for	O
maintaining	O
a	O
collection	B-Application
of	O
items	O
with	O
numerical	O
priorities	O
,	O
having	O
operations	O
for	O
finding	O
and	O
removing	O
the	O
item	O
with	O
the	O
minimum	O
priority	O
value	O
.	O
</s>
<s>
Comparison-based	O
priority	B-Application
queues	I-Application
such	O
as	O
the	O
binary	B-Application
heap	I-Application
take	O
logarithmic	O
time	O
per	O
update	O
,	O
but	O
other	O
structures	O
such	O
as	O
the	O
van	B-Application
Emde	I-Application
Boas	I-Application
tree	I-Application
or	O
bucket	B-Application
queue	I-Application
may	O
be	O
faster	O
for	O
inputs	O
whose	O
priorities	O
are	O
small	O
integers	O
.	O
</s>
<s>
These	O
data	O
structures	O
can	O
be	O
used	O
in	O
the	O
selection	B-Algorithm
sort	I-Algorithm
algorithm	O
,	O
which	O
sorts	O
a	O
collection	B-Application
of	O
elements	O
by	O
repeatedly	O
finding	O
and	O
removing	O
the	O
smallest	O
element	O
from	O
the	O
collection	B-Application
,	O
and	O
returning	O
the	O
elements	O
in	O
the	O
order	O
they	O
were	O
found	O
.	O
</s>
<s>
A	O
priority	B-Application
queue	I-Application
can	O
be	O
used	O
to	O
maintain	O
the	O
collection	B-Application
of	O
elements	O
in	O
this	O
algorithm	O
,	O
and	O
the	O
time	O
for	O
this	O
algorithm	O
on	O
a	O
collection	B-Application
of	O
elements	O
can	O
be	O
bounded	O
by	O
the	O
time	O
to	O
initialize	O
the	O
priority	B-Application
queue	I-Application
and	O
then	O
to	O
perform	O
find	O
and	O
remove	O
operations	O
.	O
</s>
<s>
For	O
instance	O
,	O
using	O
a	O
binary	B-Application
heap	I-Application
as	O
a	O
priority	B-Application
queue	I-Application
in	O
selection	B-Algorithm
sort	I-Algorithm
leads	O
to	O
the	O
heap	B-Application
sort	I-Application
algorithm	O
,	O
a	O
comparison	B-Algorithm
sorting	I-Algorithm
algorithm	O
that	O
takes	O
time	O
.	O
</s>
<s>
Instead	O
,	O
using	O
selection	B-Algorithm
sort	I-Algorithm
with	O
a	O
bucket	B-Application
queue	I-Application
gives	O
a	O
form	O
of	O
pigeonhole	B-Algorithm
sort	I-Algorithm
,	O
and	O
using	O
van	B-Application
Emde	I-Application
Boas	I-Application
trees	I-Application
or	O
other	O
integer	O
priority	B-Application
queues	I-Application
leads	O
to	O
other	O
fast	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
.	O
</s>
<s>
Instead	O
of	O
using	O
an	O
integer	O
priority	B-Application
queue	I-Application
in	O
a	O
sorting	B-Algorithm
algorithm	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
go	O
the	O
other	O
direction	O
,	O
and	O
use	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
as	O
subroutines	O
within	O
an	O
integer	O
priority	B-Application
queue	I-Application
data	O
structure	O
.	O
</s>
<s>
used	O
this	O
idea	O
to	O
show	O
that	O
,	O
if	O
it	O
is	O
possible	O
to	O
perform	O
integer	B-Algorithm
sorting	I-Algorithm
in	O
time	O
per	O
key	O
,	O
then	O
the	O
same	O
time	O
bound	O
applies	O
to	O
the	O
time	O
per	O
insertion	O
or	O
deletion	O
operation	O
in	O
a	O
priority	B-Application
queue	I-Application
data	O
structure	O
.	O
</s>
<s>
Thorup	O
's	O
reduction	O
is	O
complicated	O
and	O
assumes	O
the	O
availability	O
of	O
either	O
fast	O
multiplication	O
operations	O
or	O
table	O
lookups	O
,	O
but	O
he	O
also	O
provides	O
an	O
alternative	O
priority	B-Application
queue	I-Application
using	O
only	O
addition	O
and	O
Boolean	O
operations	O
with	O
time	O
per	O
operation	O
,	O
at	O
most	O
multiplying	O
the	O
time	O
by	O
an	O
iterated	O
logarithm	O
.	O
</s>
<s>
The	O
classical	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
of	O
pigeonhole	B-Algorithm
sort	I-Algorithm
,	O
counting	B-Algorithm
sort	I-Algorithm
,	O
and	O
radix	B-Algorithm
sort	I-Algorithm
are	O
widely	O
used	O
and	O
practical	O
.	O
</s>
<s>
experiments	O
have	O
shown	O
that	O
some	O
of	O
these	O
methods	O
may	O
be	O
an	O
improvement	O
on	O
radix	B-Algorithm
sorting	I-Algorithm
for	O
data	O
with	O
128	O
or	O
more	O
bits	O
per	O
key	O
.	O
</s>
<s>
Additionally	O
,	O
for	O
large	O
data	O
sets	O
,	O
the	O
near-random	O
memory	O
access	O
patterns	O
of	O
many	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
can	O
handicap	O
them	O
compared	O
to	O
comparison	B-Algorithm
sorting	I-Algorithm
algorithms	O
that	O
have	O
been	O
designed	O
with	O
the	O
memory	B-General_Concept
hierarchy	I-General_Concept
in	O
mind	O
.	O
</s>
<s>
Integer	B-Algorithm
sorting	I-Algorithm
provides	O
one	O
of	O
the	O
six	O
benchmarks	O
in	O
the	O
DARPA	O
High	O
Productivity	O
Computing	O
Systems	O
Discrete	O
Mathematics	O
benchmark	O
suite	O
,	O
and	O
one	O
of	O
eleven	O
benchmarks	O
in	O
the	O
NAS	B-Device
Parallel	I-Device
Benchmarks	I-Device
suite	O
.	O
</s>
<s>
Pigeonhole	B-Algorithm
sort	I-Algorithm
or	O
counting	B-Algorithm
sort	I-Algorithm
can	O
both	O
sort	O
data	O
items	O
having	O
keys	O
in	O
the	O
range	O
from	O
to	O
in	O
time	O
.	O
</s>
<s>
In	O
pigeonhole	B-Algorithm
sort	I-Algorithm
(	O
often	O
called	O
bucket	O
sort	O
)	O
,	O
pointers	O
to	O
the	O
data	O
items	O
are	O
distributed	O
to	O
a	O
table	O
of	O
buckets	O
,	O
represented	O
as	O
collection	B-Application
data	O
types	O
such	O
as	O
linked	B-Data_Structure
lists	I-Data_Structure
,	O
using	O
the	O
keys	O
as	O
indices	O
into	O
the	O
table	O
.	O
</s>
<s>
Counting	B-Algorithm
sort	I-Algorithm
uses	O
a	O
table	O
of	O
counters	O
in	O
place	O
of	O
a	O
table	O
of	O
buckets	O
,	O
to	O
determine	O
the	O
number	O
of	O
items	O
with	O
each	O
key	O
.	O
</s>
<s>
Then	O
,	O
a	O
prefix	B-Application
sum	I-Application
computation	O
is	O
used	O
to	O
determine	O
the	O
range	O
of	O
positions	O
in	O
the	O
sorted	O
output	O
at	O
which	O
the	O
values	O
with	O
each	O
key	O
should	O
be	O
placed	O
.	O
</s>
<s>
Radix	B-Algorithm
sort	I-Algorithm
is	O
a	O
sorting	B-Algorithm
algorithm	I-Algorithm
that	O
works	O
for	O
larger	O
keys	O
than	O
pigeonhole	B-Algorithm
sort	I-Algorithm
or	O
counting	B-Algorithm
sort	I-Algorithm
by	O
performing	O
multiple	O
passes	O
over	O
the	O
data	O
.	O
</s>
<s>
Each	O
pass	O
sorts	O
the	O
input	O
using	O
only	O
part	O
of	O
the	O
keys	O
,	O
by	O
using	O
a	O
different	O
sorting	B-Algorithm
algorithm	I-Algorithm
(	O
such	O
as	O
pigeonhole	B-Algorithm
sort	I-Algorithm
or	O
counting	B-Algorithm
sort	I-Algorithm
)	O
that	O
is	O
suited	O
only	O
for	O
small	O
keys	O
.	O
</s>
<s>
To	O
break	O
the	O
keys	O
into	O
parts	O
,	O
the	O
radix	B-Algorithm
sort	I-Algorithm
algorithm	O
computes	O
the	O
positional	O
notation	O
for	O
each	O
key	O
,	O
</s>
<s>
For	O
this	O
algorithm	O
to	O
work	O
correctly	O
,	O
the	O
sorting	B-Algorithm
algorithm	I-Algorithm
used	O
in	O
each	O
pass	O
over	O
the	O
data	O
must	O
be	O
stable	O
:	O
items	O
with	O
equal	O
digits	O
should	O
not	O
change	O
positions	O
with	O
each	O
other	O
.	O
</s>
<s>
With	O
these	O
choices	O
,	O
and	O
with	O
pigeonhole	B-Algorithm
sort	I-Algorithm
or	O
counting	B-Algorithm
sort	I-Algorithm
as	O
the	O
base	O
algorithm	O
,	O
the	O
radix	B-Algorithm
sorting	I-Algorithm
algorithm	O
can	O
sort	O
data	O
items	O
having	O
keys	O
in	O
the	O
range	O
from	O
to	O
in	O
time	O
.	O
</s>
<s>
However	O
,	O
despite	O
their	O
theoretical	O
advantages	O
,	O
these	O
algorithms	O
are	O
not	O
an	O
improvement	O
for	O
the	O
typical	O
ranges	O
of	O
these	O
parameters	O
that	O
arise	O
in	O
practical	O
sorting	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
A	O
Van	B-Application
Emde	I-Application
Boas	I-Application
tree	I-Application
may	O
be	O
used	O
as	O
a	O
priority	B-Application
queue	I-Application
to	O
sort	O
a	O
set	O
of	O
keys	O
,	O
each	O
in	O
the	O
range	O
from	O
to	O
,	O
in	O
time	O
.	O
</s>
<s>
This	O
is	O
a	O
theoretical	O
improvement	O
over	O
radix	B-Algorithm
sorting	I-Algorithm
when	O
is	O
sufficiently	O
large	O
.	O
</s>
<s>
However	O
,	O
in	O
order	O
to	O
use	O
a	O
Van	B-Application
Emde	I-Application
Boas	I-Application
tree	I-Application
,	O
one	O
either	O
needs	O
a	O
directly	O
addressable	O
memory	O
of	O
words	O
,	O
or	O
one	O
needs	O
to	O
simulate	O
it	O
using	O
a	O
hash	B-Algorithm
table	I-Algorithm
,	O
reducing	O
the	O
space	O
to	O
linear	O
but	O
making	O
the	O
algorithm	O
randomized	O
.	O
</s>
<s>
Another	O
priority	B-Application
queue	I-Application
with	O
similar	O
performance	O
(	O
including	O
the	O
need	O
for	O
randomization	O
in	O
the	O
form	O
of	O
hash	B-Algorithm
tables	I-Algorithm
)	O
is	O
the	O
Y-fast	B-Data_Structure
trie	I-Data_Structure
of	O
.	O
</s>
<s>
They	O
observed	O
that	O
each	O
pass	O
of	O
radix	B-Algorithm
sort	I-Algorithm
can	O
be	O
interpreted	O
as	O
a	O
range	O
reduction	O
technique	O
that	O
,	O
in	O
linear	O
time	O
,	O
reduces	O
the	O
maximum	O
key	O
size	O
by	O
a	O
factor	O
of	O
;	O
instead	O
,	O
their	O
technique	O
reduces	O
the	O
key	O
size	O
to	O
the	O
square	O
root	O
of	O
its	O
previous	O
value	O
(	O
halving	O
the	O
number	O
of	O
bits	O
needed	O
to	O
represent	O
a	O
key	O
)	O
,	O
again	O
in	O
linear	O
time	O
.	O
</s>
<s>
As	O
in	O
radix	B-Algorithm
sort	I-Algorithm
,	O
they	O
interpret	O
the	O
keys	O
as	O
two-digit	O
base	O
-	O
numbers	O
for	O
a	O
base	O
that	O
is	O
approximately	O
.	O
</s>
<s>
They	O
then	O
group	O
the	O
items	O
to	O
be	O
sorted	O
into	O
buckets	O
according	O
to	O
their	O
high	O
digits	O
,	O
in	O
linear	O
time	O
,	O
using	O
either	O
a	O
large	O
but	O
uninitialized	O
direct	O
addressed	O
memory	O
or	O
a	O
hash	B-Algorithm
table	I-Algorithm
.	O
</s>
<s>
By	O
grouping	O
the	O
items	O
from	O
this	O
list	O
into	O
buckets	O
again	O
,	O
each	O
bucket	O
may	O
be	O
placed	O
into	O
sorted	O
order	O
,	O
and	O
by	O
extracting	O
the	O
representatives	O
from	O
the	O
sorted	B-Algorithm
list	I-Algorithm
the	O
buckets	O
may	O
be	O
concatenated	O
together	O
into	O
sorted	O
order	O
.	O
</s>
<s>
Thus	O
,	O
in	O
linear	O
time	O
,	O
the	O
sorting	B-Algorithm
problem	I-Algorithm
is	O
reduced	O
to	O
another	O
recursive	O
sorting	B-Algorithm
problem	I-Algorithm
in	O
which	O
the	O
keys	O
are	O
much	O
smaller	O
,	O
the	O
square	O
root	O
of	O
their	O
previous	O
magnitude	O
.	O
</s>
<s>
A	O
complicated	O
randomized	O
algorithm	O
of	O
in	O
the	O
word	B-Application
RAM	I-Application
model	O
of	O
computation	O
allows	O
these	O
time	O
bounds	O
to	O
be	O
reduced	O
even	O
farther	O
,	O
to	O
.	O
</s>
<s>
An	O
integer	B-Algorithm
sorting	I-Algorithm
algorithm	O
is	O
said	O
to	O
be	O
non-conservative	O
if	O
it	O
requires	O
a	O
word	O
size	O
that	O
is	O
significantly	O
larger	O
than	O
.	O
</s>
<s>
As	O
an	O
extreme	O
instance	O
,	O
if	O
,	O
and	O
all	O
keys	O
are	O
distinct	O
,	O
then	O
the	O
set	O
of	O
keys	O
may	O
be	O
sorted	O
in	O
linear	O
time	O
by	O
representing	O
it	O
as	O
a	O
bitvector	B-Data_Structure
,	O
with	O
a	O
1	O
bit	O
in	O
position	O
when	O
is	O
one	O
of	O
the	O
input	O
keys	O
,	O
and	O
then	O
repeatedly	O
removing	O
the	O
least	O
significant	O
bit	O
.	O
</s>
<s>
The	O
non-conservative	O
packed	O
sorting	B-Algorithm
algorithm	I-Algorithm
of	O
uses	O
a	O
subroutine	O
,	O
based	O
on	O
Ken	O
Batcher	O
's	O
bitonic	B-Algorithm
sorting	I-Algorithm
network	I-Algorithm
,	O
for	O
merging	B-Algorithm
two	O
sorted	O
sequences	O
of	O
keys	O
that	O
are	O
each	O
short	O
enough	O
to	O
be	O
packed	O
into	O
a	O
single	O
machine	O
word	O
.	O
</s>
<s>
The	O
input	O
to	O
the	O
packed	O
sorting	B-Algorithm
algorithm	I-Algorithm
,	O
a	O
sequence	O
of	O
items	O
stored	O
one	O
per	O
word	O
,	O
is	O
transformed	O
into	O
a	O
packed	O
form	O
,	O
a	O
sequence	O
of	O
words	O
each	O
holding	O
multiple	O
items	O
in	O
sorted	O
order	O
,	O
by	O
using	O
this	O
subroutine	O
repeatedly	O
to	O
double	O
the	O
number	O
of	O
items	O
packed	O
into	O
each	O
word	O
.	O
</s>
<s>
Once	O
the	O
sequence	O
is	O
in	O
packed	O
form	O
,	O
Albers	O
and	O
Hagerup	O
use	O
a	O
form	O
of	O
merge	B-Algorithm
sort	I-Algorithm
to	O
sort	O
it	O
;	O
when	O
two	O
sequences	O
are	O
being	O
merged	O
to	O
form	O
a	O
single	O
longer	O
sequence	O
,	O
the	O
same	O
bitonic	B-Algorithm
sorting	I-Algorithm
subroutine	O
can	O
be	O
used	O
to	O
repeatedly	O
extract	O
packed	O
words	O
consisting	O
of	O
the	O
smallest	O
remaining	O
elements	O
of	O
the	O
two	O
sequences	O
.	O
</s>
<s>
Pigeonhole	B-Algorithm
sort	I-Algorithm
,	O
counting	B-Algorithm
sort	I-Algorithm
,	O
radix	B-Algorithm
sort	I-Algorithm
,	O
and	O
Van	B-Application
Emde	I-Application
Boas	I-Application
tree	I-Application
sorting	B-Algorithm
all	O
work	O
best	O
when	O
the	O
key	O
size	O
is	O
small	O
;	O
for	O
large	O
enough	O
keys	O
,	O
they	O
become	O
slower	O
than	O
comparison	B-Algorithm
sorting	I-Algorithm
algorithms	O
.	O
</s>
<s>
An	O
early	O
result	O
in	O
this	O
direction	O
was	O
provided	O
by	O
using	O
the	O
cell-probe	B-Application
model	I-Application
of	O
computation	O
(	O
an	O
artificial	O
model	O
in	O
which	O
the	O
complexity	O
of	O
an	O
algorithm	O
is	O
measured	O
only	O
by	O
the	O
number	O
of	O
memory	O
accesses	O
it	O
performs	O
)	O
.	O
</s>
<s>
Building	O
on	O
their	O
work	O
,	O
described	O
two	O
data	O
structures	O
,	O
the	O
Q-heap	O
and	O
the	O
atomic	O
heap	O
,	O
that	O
are	O
implementable	O
on	O
a	O
random	B-Application
access	I-Application
machine	I-Application
.	O
</s>
<s>
The	O
Q-heap	O
is	O
a	O
bit-parallel	O
version	O
of	O
a	O
binary	O
trie	B-General_Concept
,	O
and	O
allows	O
both	O
priority	B-Application
queue	I-Application
operations	O
and	O
successor	O
and	O
predecessor	O
queries	O
to	O
be	O
performed	O
in	O
constant	O
time	O
for	O
sets	O
of	O
items	O
,	O
where	O
is	O
the	O
size	O
of	O
the	O
precomputed	O
tables	O
needed	O
to	O
implement	O
the	O
data	O
structure	O
.	O
</s>
<s>
The	O
atomic	O
heap	O
is	O
a	O
B-tree	B-Architecture
in	O
which	O
each	O
tree	O
node	O
is	O
represented	O
as	O
a	O
Q-heap	O
;	O
it	O
allows	O
constant	O
time	O
priority	B-Application
queue	I-Application
operations	O
(	O
and	O
therefore	O
sorting	B-Algorithm
)	O
for	O
sets	O
of	O
items	O
.	O
</s>
<s>
provide	O
a	O
randomized	O
algorithm	O
called	O
signature	O
sort	O
that	O
allows	O
for	O
linear	O
time	O
sorting	B-Algorithm
of	O
sets	O
of	O
up	O
to	O
items	O
at	O
a	O
time	O
,	O
for	O
any	O
constant	O
.	O
</s>
<s>
If	O
is	O
sufficiently	O
small	O
,	O
the	O
numbers	O
formed	O
by	O
this	O
replacement	O
process	O
will	O
be	O
significantly	O
smaller	O
than	O
the	O
original	O
keys	O
,	O
allowing	O
the	O
non-conservative	O
packed	O
sorting	B-Algorithm
algorithm	I-Algorithm
of	O
to	O
sort	O
the	O
replaced	O
numbers	O
in	O
linear	O
time	O
.	O
</s>
<s>
From	O
the	O
sorted	B-Algorithm
list	I-Algorithm
of	O
replaced	O
numbers	O
,	O
it	O
is	O
possible	O
to	O
form	O
a	O
compressed	O
trie	B-General_Concept
of	O
the	O
keys	O
in	O
linear	O
time	O
,	O
and	O
the	O
children	O
of	O
each	O
node	O
in	O
the	O
trie	B-General_Concept
may	O
be	O
sorted	O
recursively	O
using	O
only	O
keys	O
of	O
size	O
,	O
after	O
which	O
a	O
tree	O
traversal	O
produces	O
the	O
sorted	O
order	O
of	O
the	O
items	O
.	O
</s>
<s>
introduced	O
the	O
transdichotomous	O
model	O
of	O
analysis	O
for	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
,	O
in	O
which	O
nothing	O
is	O
assumed	O
about	O
the	O
range	O
of	O
the	O
integer	O
keys	O
and	O
one	O
must	O
bound	O
the	O
algorithm	O
's	O
performance	O
by	O
a	O
function	O
of	O
the	O
number	O
of	O
data	O
values	O
alone	O
.	O
</s>
<s>
Alternatively	O
,	O
in	O
this	O
model	O
,	O
the	O
running	O
time	O
for	O
an	O
algorithm	O
on	O
a	O
set	O
of	O
items	O
is	O
assumed	O
to	O
be	O
the	O
worst	B-General_Concept
case	I-General_Concept
running	O
time	O
for	O
any	O
possible	O
combination	O
of	O
values	O
of	O
and	O
.	O
</s>
<s>
The	O
first	O
algorithm	O
of	O
this	O
type	O
was	O
Fredman	O
and	O
Willard	O
's	O
fusion	B-Data_Structure
tree	I-Data_Structure
sorting	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
runs	O
in	O
time	O
;	O
this	O
is	O
an	O
improvement	O
over	O
comparison	B-Algorithm
sorting	I-Algorithm
for	O
any	O
choice	O
of	O
and	O
.	O
</s>
<s>
For	O
instance	O
,	O
by	O
repeatedly	O
applying	O
the	O
Kirkpatrick	O
–	O
Reisch	O
range	O
reduction	O
technique	O
until	O
the	O
keys	O
are	O
small	O
enough	O
to	O
apply	O
the	O
Albers	O
–	O
Hagerup	O
packed	O
sorting	B-Algorithm
algorithm	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
sort	O
in	O
time	O
;	O
however	O
,	O
the	O
range	O
reduction	O
part	O
of	O
this	O
algorithm	O
requires	O
either	O
a	O
large	O
memory	O
(	O
proportional	O
to	O
)	O
or	O
randomization	O
in	O
the	O
form	O
of	O
hash	B-Algorithm
tables	I-Algorithm
.	O
</s>
<s>
Their	O
technique	O
involves	O
using	O
ideas	O
related	O
to	O
signature	O
sorting	B-Algorithm
to	O
partition	O
the	O
data	O
into	O
many	O
small	O
sublists	O
,	O
of	O
a	O
size	O
small	O
enough	O
that	O
signature	O
sorting	B-Algorithm
can	O
sort	O
each	O
of	O
them	O
efficiently	O
.	O
</s>
