<s>
In	O
parallel	B-Operating_System
algorithms	I-Operating_System
,	O
the	O
list	B-Operating_System
ranking	I-Operating_System
problem	O
involves	O
determining	O
the	O
position	O
,	O
or	O
rank	O
,	O
of	O
each	O
item	O
in	O
a	O
linked	B-Data_Structure
list	I-Data_Structure
.	O
</s>
<s>
As	O
wrote	O
,	O
the	O
problem	O
was	O
viewed	O
as	O
important	O
in	O
the	O
parallel	B-Operating_System
algorithms	I-Operating_System
community	O
both	O
for	O
its	O
many	O
applications	O
and	O
because	O
solving	O
it	O
led	O
to	O
many	O
important	O
ideas	O
that	O
could	O
be	O
applied	O
in	O
parallel	B-Operating_System
algorithms	I-Operating_System
more	O
generally	O
.	O
</s>
<s>
The	O
list	B-Operating_System
ranking	I-Operating_System
problem	O
was	O
posed	O
by	O
,	O
who	O
solved	O
it	O
with	O
a	O
parallel	B-Operating_System
algorithm	I-Operating_System
using	O
logarithmic	O
time	O
and	O
O(n log n )	O
total	O
steps	O
(	O
that	O
is	O
,	O
O(n )	O
processors	O
)	O
.	O
</s>
<s>
Over	O
a	O
sequence	O
of	O
many	O
subsequent	O
papers	O
,	O
this	O
was	O
eventually	O
improved	O
to	O
linearly	O
many	O
steps	O
(	O
O( 	O
n/log	O
n	O
)	O
processors	O
)	O
,	O
on	O
the	O
most	O
restrictive	O
model	O
of	O
synchronous	O
shared-memory	O
parallel	O
computation	O
,	O
the	O
exclusive	B-Operating_System
read	I-Operating_System
exclusive	I-Operating_System
write	I-Operating_System
PRAM	B-Operating_System
(	O
;	O
;	O
)	O
.	O
</s>
<s>
List	B-Operating_System
ranking	I-Operating_System
can	O
equivalently	O
be	O
viewed	O
as	O
performing	O
a	O
prefix	B-Application
sum	I-Application
operation	O
on	O
the	O
given	O
list	O
,	O
in	O
which	O
the	O
values	O
to	O
be	O
summed	O
are	O
all	O
equal	O
to	O
one	O
.	O
</s>
<s>
The	O
list	B-Operating_System
ranking	I-Operating_System
problem	O
can	O
be	O
used	O
to	O
solve	O
many	O
problems	O
on	O
trees	O
via	O
an	O
Euler	O
tour	O
technique	O
,	O
in	O
which	O
one	O
forms	O
a	O
linked	B-Data_Structure
list	I-Data_Structure
that	O
includes	O
two	O
copies	O
of	O
each	O
edge	O
of	O
the	O
tree	O
,	O
one	O
in	O
each	O
direction	O
,	O
places	O
the	O
nodes	O
of	O
this	O
list	O
into	O
an	O
ordered	O
array	O
using	O
list	B-Operating_System
ranking	I-Operating_System
,	O
and	O
then	O
performs	O
prefix	B-Application
sum	I-Application
computations	O
on	O
the	O
ordered	O
array	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
height	O
of	O
each	O
node	O
in	O
the	O
tree	O
may	O
be	O
computed	O
by	O
an	O
algorithm	O
of	O
this	O
type	O
in	O
which	O
the	O
prefix	B-Application
sum	I-Application
adds	O
1	O
for	O
each	O
downward	O
edge	O
and	O
subtracts	O
1	O
for	O
each	O
upward	O
edge	O
.	O
</s>
