<s>
-dimensional	O
hypercube	B-Operating_System
is	O
a	O
network	O
topology	O
for	O
parallel	O
computers	O
with	O
processing	O
elements	O
.	O
</s>
<s>
The	O
topology	O
allows	O
for	O
an	O
efficient	O
implementation	O
of	O
some	O
basic	O
communication	O
primitives	O
such	O
as	O
Broadcast	B-Operating_System
,	O
All-Reduce	O
,	O
and	O
Prefix	B-Application
sum	I-Application
.	O
</s>
<s>
In	O
the	O
beginning	O
of	O
a	O
prefix	B-Application
sum	I-Application
operation	O
,	O
each	O
processing	O
element	O
owns	O
a	O
message	O
.	O
</s>
<s>
Output	O
:	O
prefix	B-Application
sum	I-Application
of	O
processor	O
.	O
</s>
<s>
Observe	O
that	O
hypercubes	B-Operating_System
of	O
dimension	O
can	O
be	O
split	O
into	O
two	O
hypercubes	B-Operating_System
of	O
dimension	O
.	O
</s>
<s>
Once	O
both	O
sub	O
cubes	O
have	O
calculated	O
the	O
prefix	B-Application
sum	I-Application
,	O
the	O
sum	O
over	O
all	O
elements	O
in	O
the	O
0-sub	O
cube	O
has	O
to	O
be	O
added	O
to	O
the	O
every	O
element	O
in	O
the	O
1-sub	O
cube	O
,	O
since	O
every	O
processing	O
element	O
in	O
the	O
0-sub	O
cube	O
has	O
a	O
lower	O
rank	O
than	O
the	O
processing	O
elements	O
in	O
the	O
1-sub	O
cube	O
.	O
</s>
<s>
The	O
pseudo	O
code	O
stores	O
the	O
prefix	B-Application
sum	I-Application
in	O
variable	O
and	O
the	O
sum	O
over	O
all	O
nodes	O
in	O
a	O
sub	O
cube	O
in	O
variable	O
.	O
</s>
<s>
The	O
same	O
principle	O
can	O
be	O
applied	O
to	O
the	O
All-Reduce	O
operations	O
,	O
but	O
instead	O
of	O
concatenating	O
the	O
messages	O
,	O
it	O
performs	O
a	O
reduction	O
operation	O
on	O
the	O
two	O
messages	O
.	O
</s>
<s>
So	O
it	O
is	O
a	O
Reduce	B-Operating_System
operation	O
,	O
where	O
all	O
processing	O
units	O
know	O
the	O
result	O
.	O
</s>
<s>
Compared	O
to	O
a	O
normal	O
reduce	B-Operating_System
operation	O
followed	O
by	O
a	O
broadcast	B-Operating_System
,	O
All-Reduce	O
in	O
hypercubes	B-Operating_System
reduces	O
the	O
number	O
of	O
communication	O
steps	O
.	O
</s>
<s>
The	O
ESBT-broadcast	O
(	O
Edge-disjoint	O
Spanning	O
Binomial	O
Tree	O
)	O
algorithm	O
is	O
a	O
pipelined	O
broadcast	B-Operating_System
algorithm	O
with	O
optimal	O
runtime	O
for	O
clusters	O
with	O
hypercube	B-Operating_System
network	O
topology	O
.	O
</s>
<s>
The	O
algorithm	O
embeds	O
edge-disjoint	O
binomial	O
trees	O
in	O
the	O
hypercube	B-Operating_System
,	O
such	O
that	O
each	O
neighbor	O
of	O
processing	O
element	O
is	O
the	O
root	O
of	O
a	O
spanning	O
binomial	O
tree	O
on	O
nodes	O
.	O
</s>
<s>
To	O
broadcast	B-Operating_System
a	O
message	O
,	O
the	O
source	O
node	O
splits	O
its	O
message	O
into	O
chunks	O
of	O
equal	O
size	O
and	O
cyclically	O
sends	O
them	O
to	O
the	O
roots	O
of	O
the	O
binomial	O
trees	O
.	O
</s>
<s>
Upon	O
receiving	O
a	O
chunk	O
,	O
the	O
binomial	O
trees	O
broadcast	B-Operating_System
it	O
.	O
</s>
<s>
Broadcasting	B-Operating_System
the	O
chunk	O
within	O
the	O
binomial	O
tree	O
takes	O
steps	O
.	O
</s>
<s>
Thus	O
,	O
it	O
takes	O
steps	O
to	O
distribute	O
all	O
chunks	O
and	O
additionally	O
steps	O
until	O
the	O
last	O
binomial	O
tree	O
broadcast	B-Operating_System
has	O
finished	O
,	O
resulting	O
in	O
steps	O
overall	O
.	O
</s>
<s>
The	O
resulting	O
binomial	O
trees	O
are	O
edge-disjoint	O
and	O
therefore	O
fulfill	O
the	O
requirements	O
for	O
the	O
ESBT-broadcasting	O
algorithm	O
.	O
</s>
