<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
reduction	B-Operating_System
operator	I-Operating_System
is	O
a	O
type	O
of	O
operator	O
that	O
is	O
commonly	O
used	O
in	O
parallel	B-Operating_System
programming	I-Operating_System
to	O
reduce	O
the	O
elements	O
of	O
an	O
array	O
into	O
a	O
single	O
result	O
.	O
</s>
<s>
Reduction	B-Operating_System
operators	I-Operating_System
are	O
associative	O
and	O
often	O
(	O
but	O
not	O
necessarily	O
)	O
commutative	O
.	O
</s>
<s>
The	O
reduction	O
of	O
sets	O
of	O
elements	O
is	O
an	O
integral	O
part	O
of	O
programming	O
models	O
such	O
as	O
Map	B-Operating_System
Reduce	I-Operating_System
,	O
where	O
a	O
reduction	B-Operating_System
operator	I-Operating_System
is	O
applied	O
(	O
mapped	O
)	O
to	O
all	O
elements	O
before	O
they	O
are	O
reduced	O
.	O
</s>
<s>
Other	O
parallel	B-Operating_System
algorithms	I-Operating_System
use	O
reduction	B-Operating_System
operators	I-Operating_System
as	O
primary	O
operations	O
to	O
solve	O
more	O
complex	O
problems	O
.	O
</s>
<s>
Many	O
reduction	B-Operating_System
operators	I-Operating_System
can	O
be	O
used	O
for	O
broadcasting	O
to	O
distribute	O
data	O
to	O
all	O
processors	O
.	O
</s>
<s>
A	O
reduction	B-Operating_System
operator	I-Operating_System
can	O
help	O
break	O
down	O
a	O
task	O
into	O
various	O
partial	O
tasks	O
by	O
calculating	O
partial	O
results	O
which	O
can	O
be	O
used	O
to	O
obtain	O
a	O
final	O
result	O
.	O
</s>
<s>
A	O
reduction	B-Operating_System
operator	I-Operating_System
stores	O
the	O
result	O
of	O
the	O
partial	O
tasks	O
into	O
a	O
private	O
copy	O
of	O
the	O
variable	O
.	O
</s>
<s>
An	O
operator	O
is	O
a	O
reduction	B-Operating_System
operator	I-Operating_System
if	O
:	O
</s>
<s>
A	O
reduction	B-Operating_System
operator	I-Operating_System
can	O
be	O
applied	O
in	O
constant	O
time	O
on	O
an	O
input	O
set	O
of	O
vectors	O
with	O
elements	O
each	O
.	O
</s>
<s>
Sequential	O
algorithms	O
can	O
not	O
perform	O
better	O
than	O
linear	O
time	O
,	O
but	O
parallel	B-Operating_System
algorithms	I-Operating_System
leave	O
some	O
space	O
left	O
to	O
optimize	O
.	O
</s>
<s>
Since	O
'	O
+	O
 '	O
is	O
both	O
commutative	O
and	O
associative	O
,	O
it	O
is	O
a	O
reduction	B-Operating_System
operator	I-Operating_System
.	O
</s>
<s>
Therefore	O
this	O
reduction	O
can	O
be	O
performed	O
in	O
parallel	O
using	O
several	O
cores	O
,	O
where	O
each	O
core	O
computes	O
the	O
sum	O
of	O
a	O
subset	O
of	O
the	O
array	O
,	O
and	O
the	O
reduction	B-Operating_System
operator	I-Operating_System
merges	O
the	O
results	O
.	O
</s>
<s>
Of	O
course	O
the	O
result	O
is	O
the	O
same	O
,	O
but	O
only	O
because	O
of	O
the	O
associativity	O
of	O
the	O
reduction	B-Operating_System
operator	I-Operating_System
.	O
</s>
<s>
The	O
commutativity	O
of	O
the	O
reduction	B-Operating_System
operator	I-Operating_System
would	O
be	O
important	O
if	O
there	O
were	O
a	O
master	O
core	O
distributing	O
work	O
to	O
several	O
processors	O
,	O
since	O
then	O
the	O
results	O
could	O
arrive	O
back	O
to	O
the	O
master	O
processor	O
in	O
any	O
order	O
.	O
</s>
<s>
Matrix	O
multiplication	O
is	O
not	O
a	O
reduction	B-Operating_System
operator	I-Operating_System
since	O
the	O
operation	O
is	O
not	O
commutative	O
.	O
</s>
<s>
Regarding	O
parallel	B-Operating_System
algorithms	I-Operating_System
,	O
there	O
are	O
two	O
main	O
models	O
of	O
parallel	B-Operating_System
computation	I-Operating_System
,	O
the	O
parallel	B-Operating_System
random	I-Operating_System
access	I-Operating_System
machine	I-Operating_System
as	O
an	O
extension	O
of	O
the	O
RAM	O
with	O
shared	O
memory	O
between	O
processing	O
units	O
and	O
the	O
bulk	B-Application
synchronous	I-Application
parallel	I-Application
computer	I-Application
which	O
takes	O
communication	O
and	O
synchronization	O
into	O
account	O
.	O
</s>
<s>
The	O
strategy	O
for	O
handling	O
read	O
and	O
write	O
conflicts	O
can	O
be	O
chosen	O
as	O
restrictive	O
as	O
an	O
exclusive	O
read	O
and	O
exclusive	O
write	O
(	O
EREW	B-Operating_System
)	O
.	O
</s>
<s>
The	O
speedup	O
of	O
the	O
algorithm	O
is	O
and	O
therefore	O
the	O
efficiency	B-Operating_System
is	O
.	O
</s>
<s>
The	O
efficiency	B-Operating_System
suffers	O
because	O
half	O
of	O
the	O
active	O
processing	O
units	O
become	O
inactive	O
after	O
each	O
step	O
,	O
so	O
units	O
are	O
active	O
in	O
step	O
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
PRAM-algorithm	O
,	O
in	O
the	O
distributed	B-Operating_System
memory	I-Operating_System
model	O
,	O
memory	O
is	O
not	O
shared	O
between	O
processing	O
units	O
and	O
data	O
has	O
to	O
be	O
exchanged	O
explicitly	O
between	O
processing	O
units	O
.	O
</s>
<s>
For	O
distributed	B-Operating_System
memory	I-Operating_System
models	O
,	O
it	O
can	O
make	O
sense	O
to	O
use	O
pipelined	O
communication	O
.	O
</s>
<s>
Usually	O
,	O
linear	B-General_Concept
pipelines	I-General_Concept
split	O
data	O
or	O
a	O
tasks	O
into	O
smaller	O
pieces	O
and	O
process	O
them	O
in	O
stages	O
.	O
</s>
<s>
Reduction	O
is	O
one	O
of	O
the	O
main	O
collective	B-Operating_System
operations	I-Operating_System
implemented	O
in	O
the	O
Message	B-Application
Passing	I-Application
Interface	I-Application
,	O
where	O
performance	O
of	O
the	O
used	O
algorithm	O
is	O
important	O
and	O
evaluated	O
constantly	O
for	O
different	O
use	O
cases	O
.	O
</s>
<s>
MapReduce	B-Operating_System
relies	O
heavily	O
on	O
efficient	O
reduction	O
algorithms	O
to	O
process	O
big	O
data	O
sets	O
,	O
even	O
on	O
huge	O
clusters	O
.	O
</s>
<s>
Some	O
parallel	O
sorting	B-Algorithm
algorithms	I-Algorithm
use	O
reductions	O
to	O
be	O
able	O
to	O
handle	O
very	O
big	O
data	O
sets	O
.	O
</s>
