The Xor embedding: An embedding of hypercubes onto rings and toruses
Abstract
Many parallel algorithms use hypercubes as the communication topology among processes, which make them suitable to be executed on a hypercube multicomputer. In this way the communication cost is kept to a minimum since processes can be allocated to processors in such a way that only communication between neighbor processors is required. However, the scalability of hypercube multicomputer is constrained by the fact that the interconnection cost per node increases with the total number of nodes. From the point of view of scalability, meshes and toruses are a more interesting class of interconnection topologies. In this paper the authors propose an embedding of hypercubes onto toruses of any given dimension, incuding one-dimensional toruses which are also called rings. They also prove that the embedding is optimal in the sense that it minimizes the execution time on a ring of a class of parallel algorithms frequently found in real applications, such as FFT and some class of sorting algorithms.
Full text
The
Xo
Embedding:
An Embedding
o
Hype cubes
on o
Rings and To uses
An onio
Gondez
and
Miguel Vale o-Ga cia
Uni e si a Poli &cnica de Ca alunya
Depa amen
d'
A qui ec u a de Compu ado s
c/
G an
Capi an dn, Campus
Nod
-
Edi ici
D6
BO8071
Ba celona (Spain)
E-mail: (an onio,miguel} 9ac.upc.e~
Abs ac
Many pa allel algo i hms use hype cubes
as
he communica ion opology anwngp ocesses, which
make
hem sui able o be execu ed
on
a hype cube mul icompu e .
In
his way he communica ion
cos is kep
minimum
since p ocesses can
be
alloca ed o p ocesso s in such a way ha only
communica ion be ween neighbo p ocesso s is equi ed. Howe e , he scalabili y o hype cube
mul icompu e
is
cons ained
by
he ac ha he in e connec ion cos
pe
node
inc eases wi h he
o al numbe
o
nodes.
F om he poin o iew o scalabili y, meshes and o uses a e a mo e
in e es ing class
o
in e connec ion opologies.
In
his pape we p opose an embedding o
hype cubes
on o
omes
o
any gi en dimension, including one-dimensional omes which a e also
called ings. We
also
p o e ha his embedding
is
op imal in he sense ha i minimizes he
execu ion ime
on
a ing o a class o pa allel algo i hms equen ly
ound
in eal applica ions, such
as
FFT and
some
class o so ing algo i hms.
1.
In oduc ion
Many pa allel algo i hms use hype cubes
as
he
communica ion opology among p ocesses.
Some examples include pa allel algo i hms
o
FIT,
so s,
e c.
[Z].
We
will
call
hese
algo i hms
hype cube algo i hms
o
d-cube algo i hms.
whe e
d
is
he
numbe o dimensions
o
he hype cube.
A
hype cube algo i hm
o
dimension
d
o
d-cube algo i hm, consis s
o
2d
p ocesses
labeled om
0
o
9-1
such ha
e e y
p ocess communica es only wi h i s
d
neighbo s, one in
each
dimension
o
he d-cube.
In
his
pape we conside
he
p oblem
o
execu ing d-cube algo i hms on
mul icompu e s
[l
I.
A
mul icompu e is a dis ibu ed memo y mul ip ocesso in which
he
nodes
(p ocesso
+
local
memo y)
a e
in e connec ed h ough poin
o
poin
links.
The nodes
o
a mul icompu e
a e
in e connec ed acco ding
o
a gi en pa em
o
in e connec ion
opology.
I
his
opology is a hype cube
o
dimension
d
(d-cube mul icompu e ) hen
he
d-cube
algo i hm
can
be execu ed
on
he
mul icompu e in such
a
way ha neighbo
p ocesses
a e
mapped
on o
adjacen
nodes
(nodes di ec ly
connec ed
h ough a poin
o
poin link). We say, in
his
Case.
ha each
p ocess
o
he d-cube algo i hm
has
all i s
d
neighbo s a dis ance
I
in
he
mul icompu e
(i.e., all
equi ed
communica ion
is
be ween neighbo
nodes).
In
his
way, he cos
o
he
communica ion componen
o
he
d-cube algo i hm when i is execu ed on
a
hype cube
mul icompu e is kep minimum.
0-8186-3492-8193
$3.00
0
1993
IEEE
15
16
In e na ional
Con e ence
on Applica ion-Speci ic A ay
kesso s
An
impo an
d awback o hype cube
as
in e connec ion opology o mul lcompu en is
ha
i
is
no scalable.
In
a
dabe
mul icompu e
each
o
he
'
mdes
is
di ec ly
COM-
o
o he
d
nodes
h ough poin
o
poin links.
'Ihe e o e,
he.
cos
(and
he
complexi y) o
he
in e connec ion
ha dwa e
pe
node
inc eases wi h
he
numbex o
nodes.
0-
in e connec ion opologies, such
as
meshes
o
omes
a e
conside ed
mo e sui able o mul icompu e s wi h a la ge
numbe
o
nodes,
since
he
in e connec iOn cos
pe
node
does
no
depend
on
he
o al
numbe
o
nodes.
As
an
example,
each
node
o
il
wo-dimensional
o us
mul icompu e
is
di ec ly
co~ec ed
o
4
nodes.
i
does
no
ma e
he
numbe
o
nodes
o he mul icompu e .
When a dabe algo i hm is execu ed
on
a mul icompu e wi h a opology
o ie
han
hype cube
i is no possible, in
gene al.
o
alloca e
p ocesses
o
nodes
in such a way
ha
e e y
p ocess
has
i s
d
neighbo s
a
dis ance
1
in
he
mul icompu e .
As
an
example. in a wo-dimensional
om
mul icompu e . e e y
p ocess
has
a
mos
4
o
i s
d
neighbo s
a
dis ance
1.
I mus
ha e
a
leas
d-4
neighbo s
a
a dis ance g ea e
han
1.
A
message
o
any o
hese
" a " neighbo s mus
be
ou ed
h ough
he
poin
o
poin links
and
mdes
which
a e
ound along
he
pa h
o
he
des ina ion
node.
A
good
mapping o a d-cube algo i hm
on o
a mul icompu e will
y
o
keep
he
neighbo
p ocesses
as
close
as
possible in
he
mul icompu e , minimizing in
his
way
he
cos o he communica ion
componen o
he
d-cube algo i hm when i is execu ed
on
he
mul icompu e .
In
his
papex
we
p opose
an
embedding
o
d-cube algo i hms
on o
o us
mul icompu e s o any
a bi a y dimension.
This
embedding has
he
ollowing
p oees:
a) Fo
each
dimension o
he
d-cube, e e y
p ocess
has i s
co esponding
neighbo
a
he
same
dis ance (al hough
hese
dis ances
may
be
di e en
o
di e en dimensions).
b)
The
a e age dis ance
o
he
d
neighbo s
o
any
p ocess
is minimum, subjec
o
(a).
As
i will
be
shown
la e in
his
pape ,
p om
(a)
is
desi able
o
some kind
o
d-cube
algo i hms like
m,
since
i
gua an ees
ha
any p ocess will ne e ha e
o
wai o a neighbo
o
inish i s wo k be o e s a ing
he
in e change o
da a
wi h
i .
ROW
(b)
implies a minimum cos
in
he
communica ion componen o he d-cube algo i hm,
and
he e o e, a minimum execu ion
ime. We p esen a p oo
o
his
p ope y
o
ings
(onedimensional
o~~e~).
The
gene aliza ion
o
he p oo o
an
a bi a y dimension is s ill unde de elopmen
This
papa is
o ganized
as
ollows.
In
sec ion
2
we in oduce
some
no a ion
and
desc ibe
mo e
p ecisely
he
con ibu ion o
his
pape
as
well
as
some ela ed wo k.
Sec ions
3
p esen s he
p oposed
embedding o
dab
on o
ings.
In
sendion
4.
he
embedding is gene alized
o
omes
o
any a bi a y dimension.
In
sec ion
5
we p o e
ha
he
p oposed
embedding o ings is op imal
acco ding
o
he
c i e ion
in oduced in
sec ion
2.
Finally, we p esen
some
concluding ema ks.
2.
P elimina ies and ela ed
wo k
2.1.
De ini ions
A
d-cube
algo i hm
is
a pa allel algo i hm
ha
consis s o
2d
p ocesses
such ha e e y p ocess
communica es wi h exac ly
o he
d
p ocesses.
Tkse
d
p ocesses
a e
called i s neighbo s. We also
say
ha
he
communica ion opolo o
he
algo i hm is a hype cube. lha
means
ha he
2d
communica e) i
he
bina y
codes
o
n
and
m
di e in
a
single bi .
I
his
bi is
he
i- h bi hen
m
is
he
neighbo o
n
in dimemion i,
and
n
is
he
neighbo
o
m
in he same dimension
Then,
we w i e:
m
=
Ndn)
n
=
Ndm)
p ocesses
can
be
labeled om
0
o
Y
-1
in such a way ha p ocesses
n
andm
a e
neighbo (i.e.
hey
Session
2:
E icien
Design
Me hods
I
17
In
his
pape we ocus on d-cube
algo i hms
in which e e y
p ocess
has
he
ollowing s uc u e:
do
i=O,d-1
compu e
communica e wi h neighbo in dimension
i
enddo
In
his
algo i hm e e y
p ocess
consis s o d s ages,
each
o hem composed o a compu a ion
and
a communica ion
phase.
In
each s age, e e y
p ocess
uses
a
di e en dimension
o
exchange
in o ma ion wi h one o i s neighbo s.
We assume
ha
he du a ion o he compu e phase and he amoun o in o ma ion
o
be
exchanged is he same o all he
s ages
and all
he
p ocesses
o he d-cube algo i hm.
A
d-cube
algo i hm wi h he abo e ea u es will be called a compu e-and-communica e d-cube algo i hm,
o
a
CC
d-cube algo i hm o
sho .
'Ihis
kind
o d-cube algo i hms a e common in
eal
applica ions
like
FIT,
some
ype
o
so s,
e c.
Any pa allel algo i hm can be modelled
as
a g aph. The e ices o he g aph ep esen
he
p ocesses
o
he
algo i hm and he edges
o
he
g aph ep esen he neighbo ela ionship among
p ocesses.
A
mul icompu e can
also
be
modelled by a g aph.
The
e ices o he g aph ep eseni
he nodes
o
he mul icompu e
and
he edges o he g aph ep esen he poin
o
poin
links
which
in e connec hese nodes.
Mul icompu e s can
be
classi ied acco ding
o
hei
in e connec ion opology.
In
his
pape , we
a e in e es ed in mesh and
oms
mul icompu e s, since hey ha e scalable in e connec ion
opologies.
A
(kl,k2,
...,
kc)
c-dimensional
om
is an undi ec ed g aph in which
he
nodes
can
be labeled
as
c- uples (i1,iB
...,
ic),
0
ij
<
ki.
E e y node (i&
,....
ic)o
he
g aph has wo neighbo s in each
dimension o
he
oms.
I s
le neighbo in dimension
j
is (il
,...,($-
1)
mod
k,
...,
ic)
and
i s igh
neighbo in
his
dimension is (il,
...,
($+l)
mod
k,
...,
ic).
A
(k&,
...,
k,)
c-dimensional
mesh
is an undi ec ed g aph in which he nodes can
be
labeled
as
c- uples (i1.i~.
....
i,),
0
I
ij
e
kk
E e y node o
he
g aph has
wo
neighbo s in
each
dimension]
o
he mesh i
0
<
4
<
k,-1.
I s le neighbo is (il
....,
5-1
,...,
ic) and i s igh neighbo is (il,
...,
$+l
,...,
i,).
I
$=O.
he node
has
only a igh neighbo and i
$=k+
hen
i
only
has
a
le
neighbo .
A
line is a one-dimensional mesh while a one-dimensional o us is
called
a ing.
Figu e
1
shows some examples and illus a es how hei
nodes
a e labeled.
The dis ance in a g aph be ween wo e ices is he
minimum
numbe o edges ha join hose
e ices. In
he
pa icula case o he g aph which models a d-cube, he dis ance be ween
wo
e ices
is
known
as
he
Hamming dis ance (numbe o di e en bi s in hei bina y
ep esen a ions).
An
embedding o g aph
G
in o g aph
H
is a bijec i e unc ion
om he e ices o G
o
he
e ices o
H.
We assume
ha
G
and
H
ha e he same numbe o e ices.
The
p oblem o execu ing a
CC
d-cube algo i hm on a mul icompu e can
be
modelled
as
he
embedding
0-
g aph
G,
which ep esen s he
CC
d-cube algo i hm,
on o
g aph
H,
which ep esen s
he
mul icompu e .
The
dila ion o an edge
(n,m)
o
G
(edge joining e ices
n
and
m)
is he dis ance in
H
be ween
. W
andJ7m).
I
G
models a
CC
d-cube algo i hm, an edge
exis s
be ween e ices
n
and
m
i
m=Nin),
o
some
i
E
[O,d-I]. The dila ion o
his
edge will
be
deno ed by
Din).
Ob iously,
since
n=
Nim),
Din)
=
Dim).
When a
CC
d-cube algo i hm is execu ed on a mul icompu e ,
as
de ined by a gi en
18
In e na ional
Co& ence
on
Applica ion-Speci ic
A ay
heeo s
(4
(dl
d)
(4.4)
o us.The
pic un
also
show0
hoaw
ho
nod..
am
labeld.
Figu e
1:
Di e en
ypes
o
mul icompu e s:
a)
line,
b)
ing,
c)
(44)
mssh
and
embedding
;
a
communica ion be ween
p ocesses
n
and Nkn)
( equi ed
in i e a ion
i
o
he
CC
d-cube algo i hm)
is
implemen ed by
a
message which is ou ed h ough
Di(n)
poin
o
poin
links
and
D.W-1
nodes
o
he
mul icompu e ep esen ed by
H,
which
a e
ound in
he
sho es
pa h
be ween
nodes ln)
and lNkn)).
We
deno e
by
To
he du a ion o he a i hme ic compu a ion phase in e e y s age o
he
CC
d-cube algo i hm, when i is execu ed
on
he a ge mul icompu e . We deno e by
Tc
he cos o
sending
a
message
h ough
a
poin
o
poin
link
on
he
mul icompu e .
The ime
o
execu e
a
CC
d-cube algo i hm
on
a
mul icompu e wi h
9
nodes,
as
de ined by
embedding can
be
exp essed
as:
T
=
dTa
+
Tc
whe e
Tc is
he
cos o he communica ion componen o
he
CC
d-cube algo i hm.
Tc
can be
exp kgsed
as
ollows:
Tg
=
nuu
(Td-I(n):
n=0..2%)
Tiin) =D,in) Tc
+
nuu
(Ti-dn),
Ti.l(Nkn))
I
i"&)
=
0
(a)
(b)
In
he abo e exp essions,
Tin)
is he cos o he communica ion componen o
p ocess
n
&om
he beginning o he execu ion o he end o s age
i.
Exp ession
(a)
indica es
ha
Tgis
equal o he
highes communica ion componen
cos
o any
p ocess
a
he
end
o
he
d
s ages
o
he
CC
d-cube
algo i hm. Exp ession
(b)
gi es
he
communica ion
componen cos o
p ocess
n
a
he
end o s age
i.
In
his
s age
p ocess
n
mus exchange in o ma ion wi h i s
neighbo
Nin).
?he
cos
o exchanging
his
in o ma ion
is
Di(n)T,
Howe e ,
his
exchange
caauo
s a
un il
bo h
p ocaws
n
and
Ni(n)
a e
eady
o
do
i .
In
gene al, ei he
p ocess
n
o
p ocess
Ndn) will ha e
o
wai o
i s
neighbo
o
a i e o
he
poin in which communica ion can be
s a ed.
lMs
is
why
he
e m
"max"
appea s
in
expsion (b).
In
he ollowing, we
call
hose
idle
pe iods
as
wai ing
pe iods.
Ob iously, i he mul icompu e
has
a
d-cube
in e coIlnec ion
opology
hen
he bes embedding
is ln)
=
n
(iden i y
embedding).
In
his
case
Din)
=
I
( o e e y
i
and
n)
and he execu ion ime
is
T,
=
d(Ta+T,)
Session
2
E icien Design Me&&
I
19
2.2.
Con ibu ions
In
his
pape . we a e in e es ed in hose
embeddings
in which
Dkn)
=
Di
(i
E
[O,d-I]
and
n
E
[O,2d-I]).
This
means ha e e y
p ocess
has i s
neighbo
in dimension
i
a
he
same dis ance in
@e
a ge mul icompu e .
In
he
ollowing,
an
embedding
wi h
his
ea u e
is
called
embedding
wi h
cons un
dis ances
and
he
alues o
Di
(i
E
[O,d-I])
a e
called
he
dis unces
o he embedding.
Embeddings wi h cons an dis ances ha e
he
p ope y
ha e eay
p ocess
akes
he
same ime
o
communica e
in
any gi en s age o he
CC
d-cube algo i hm.
Because
he
du a ion o he compu e
phase
is
also
he
same
o
e e y
p ocess,
wai ing pe iods
a e
a oided since
neighbo
p ocesses
a i e
a
he
same ime
o
he poin a which hey ha e
o
communica e.
The
ime
o
execu e a
CC
d-cube algo i hm
on o
a mul icompu e .
as
de ined
by
an
embedding
wi h cons an dis ances
is:
d-
1
d-
1
T,
=
x
(Ta+DiTc)
=
dTa+TcxDi
=
d(T,+TcDa)
i=O
i=O
whe e
Dais
he
a e age dis ance
o
he
embedding:
d-
1
XDi
=
a e agedis ance
( ~
0,
-
i=O
d
The e o e.
he
embedding wi h
cons an
dis ances which
minimizes
T is
ha
whose a e age
dis ance
D,
is minimum.
An
embedding wi h such
p ope y
is
said
o
be
op imal.
As
i was men ioned in
he
in oduc ion, we
a e
in e es ed in execu ing
CC
d-cube algo i hms
on
scalable mul icompu e s.
In
pa icula . we a e in e es ed in
om
mul icompu e s
since
o meshes
a well-known embedding. desc ibed in
he
nex
sec ion,
is op imal o
CC
d-cube algo i hms.
In
his
pape
we p opose
an
embedding wi h cons an dis ances o
CC
d-cube algo i hms on o
om
mul icompu e s o any a bi a y dimension. Mo eo e , we p o e ha
he
p oposed
embedding wi h
cons an
dis ances
is op imal
o
ings (onedimensional
ow).
Ano he
addi ional
p ope y
o he
p oposed
embedding is i s simplici y, which means a negligible cos
o
compu e
he
loca ion o any
p ocess
in he mul icompu e .
2.3.
Rela edwo k
The
p oblem o embedding d-cubes on o meshes
and
o uses
has
been
p e iously conside ed
by
o hex
au ho s.
We
mw e iew some ela ed wo k.
Ma ic
p esen s
in
[7]
a s udy o
he
s anda d
embedding
(de ined below) o d-cubes
on o
wo-dimensional meshes and
o uses.
To
de ine
he
s anda d
embedding
(which will be deno ed by
$
o a d-cube
on o
a
line
o
a
ing.
he
nodes
o
he
a ge mul icompu e
a e
numbe ed om
0
o
-I
(see
igu es 1.a
and
1.b).
Then,
he
s anda d
embedding is
de ined
by
(see
igu e 2.a):
,,d(n)
=
?I
n e
s anda d
embedding o a d-cube
on o
a
(kl,k
=....kc)
cdimensional mesh
o
om
is de ined
as
ollows:
s,d(")
=
(PI*P ,
...,Pc)
20
In e na ional Con e ence on Applica ion-Speci ic A ay
P ocesso s
0123
e------*
12 13 14
01 234561
(a)
(b)
Figu e
2:
S anda d embeddings
o :
a)
a
3-c-
on o
a
line
o
a
ing
and
b)
a
kube on o
a
(4,4)
mesh
o o us.
Each
label indka es
whkh
e ex
o
he
dcube is mapped on o
each
node
o
he
mul icompu e .
W apa ound links
a e
no
shown
o
cla i y.
whe e:
i-
1
p,
=
(n
mod ikj)di nkj
Figu e 2.b shows an example in which
c=2
and
kl=k2=4.
Ob iousl
he
s anda d embedding is
a
cons an dis ance embedding. Fo he pa icula
case
in which
ki=g‘,
i
E[~,c].
he
dis ances o
he s anda d embedding
a e:
j=1
j=l
Q
=
2
i
ie[O,
d
-
I]
I can
be
shown
ha
he
s anda d
embedding is
op lmal
o
meshes.
in
he
sense
ha
i
minimizes
he a e age dis ance [3],[8]. Howe e , i is no op imal o
o uses,
as
i will be shown
la e
in
his
Pape .
Ha pe
in [41 and
Lai
and Spague in [5] sol e he p oblem o embedding d-cubes on o meshes
o
minimize
he dila ion o
he
embedding ( he maximum dila ion o any edge). Bo h
p oposals
use
he
byweigh
embedding, deno ed by h, which is no an embedding wi h cons an dis ances. Nex ,
we desc ibe b ie ly
his
embedding.
In
he
case o a line, he labels o he e ices which ep esen he
p ocesses
o he d-cube
algo i hm a e o de ed by hei weigh s.
The
weigh o a label is
he
numbe o
I’s
in
i s bina y
ep esen a ion. Labels wi h
he
same weigh
a e
o de ed
in
descending
o de .
Then,
he
p ocesses
o he d-cube o de ed in ha way a e alloca ed
o
he
nodes
o
he
line,
om
le
o
igh . Figu e 3.a
shows an example. The byweigh embedding can
be
ex ended
o
meshes o any dimension. In
pa icula , Lai and Spague ex end
his
embedding
o
wo-dimensional meshes
in
[5]. Figu e 3.b
shows an example.
The byweigh embedding
minimizes
he dila ion o
he
embedding.
?his
is
an
in e es ing
p ope y
in
some pa icula applica ions o embeddings. Fo ins ance,
Lai
and Spague p opose
his
embedding
o
sol e he p oblem o placing
he
p ocesso s o a hype cube on a p in ed ci cui boa d
o
a chip (which
can
be
modelled
as
a
wo-dimensional mesh). Howe e , he byweigh embedding
is no an embedding wi h cons an dis ances, which is an impo an p ope y
in
he con ex o
execu ing
CC
d-cube algo i hms on o mul icompu e s. ’Ihe e o e, du ing he execu ion o
he
CC
d-cube algo i hm, wai ing
pe iods
will
appu
which con ibu e
o
inc ease
he
execu ion
ime.
To
illus a e
his
ac , igu e4 shows an example in which
he
execu ion imes o a
CC
3-cube algo i hm
Session
2:
E icien
Design
Me hods
I
21
OOO
100
010
001
110
101
011
111
(a)
Figu e
3
Byweigh embeddings
o :
a)
a
3-cube
on o
a
line and b)
a
Scube
on o
a
(8,4)
mesh.
on a line
o
bo h he s anda d embedding
and
he byweigh embedding
a e
compa ed. The wai ing
pe iods
which con ibu e o
make
he byweigh embedding un slowe
han
he
s anda d embedding
a e also
shown.
In
[6],
Y
.W. Ma and
L.
Tao
p oposed se e al embeddings among
o uses
and
meshes
o di e en
dimensions.
Thei
p oposals
a e
based on gene alizing he
concep s
o g ay code o adix-2
numbe ing sys em o mix- adix numbe ing sys ems. Since a d-cube
can
also
be
seen
as
a
d-dimensional mesh
o
o us wi h wo elemen s in
each
dimension.
lki
embedding can also be
applied o sol e he p oblem add essed in
his
pape . Howe e , hey ocus on minimizing he
dila ion
( he
la ges dis ance be ween
any
wo
neighbo s o
he
d-cube) and he e o e he esul ing
embeddings in gene al do no ha e cons an dis ances, which
is
a desi able p ope y
o
ou
objec i e. Howe e , i one
s a s
wi h a d-cube ep esen ed by means
o
a (2.2,
...
2)
d-dimensional
mesh
o
o us, hen
he
esul ing embedding on o a ing
o
a wo-dimensional o us has cons an
dis ances. Ne e heless, i s a e age dis ance and he e o e i s
pex onnance
o execu ing ou a ge
algo i hm is wo se han
he
embedding p oposed in
his
pape .
3.
We desc ibe now he
p oposed
embedding wi h cons an dis ances o a d-cube on o a ing wi h
Zd
e ices (we assumed
>
I).
We call
his
embedding
xo
embedding
and i is deno ed
by ,,
The
xo embedding is op imal, in he sense ha i
minimizes
he
a e age
dis ance.
The xo embedding
is desc ibed nex . Sec ion
5
p esen s a p oo
o
i s op imali y.
Le
G
be
he g aph which ep esen s he
CC
d-cube algo i hm and
R
be he g aph which
e esen s he ing mul icompu e . Assume ha
he
e ices o
R
a e labeled om
0
o
2
-1clockwise
(see
igu e 1.b). Le
(nd-1, nd.2
...,nl,
no)
be
he
label
(in
bina y
code)
o e ex
n
in
G.
This
e ex is mapped on o e ex
m= ,,(n)
in
R,
whose label in bina y
code
(md+..,ma)
is:
An
embedding
wi h
cons an dis ances
o
ings
F
mi
=
ni
i
E
[O,
d-11,
i
#
d-2
md.2
=
XOR
(nd.19 nd.3
whe e
XOR
(qb)
is he exclusi e-o
o
bi s
a
and
b.
Figu e
5
shows an example
o
d=4.
The dis ances
o
he xo embedding a e:
Di
=
2'
i
E
[O,
d-21
Dd-1
=
2d-2
22
In e na ional Con e ence on Applica ion-Speci ic A ay
P ocesso s
124 332
124 422
124 431
124 431
124 422
124 332
124 321
Compu a ion
Communica ion
Wai ingpe iod
bw
(b)
(C)
Figu e
4
a) Dila ions
o
he s anda d and byweigh embeddings
(M).
Execu ing
a
CC
3-cube algo i hm on a line using: b) he s anda d
embudding
and
c) he byweigh embedding.
me e o e,
he
a e age
dis ance
is:
d-2
4.
An embedding
wi h
cons an dis ances
o
c-dimensional
o uses
Now we desc ibe
he
xo
embedding o a d-dimensional hype cube on o a
(2d',2d2,...,2d3
c-dimensional
o us
such ha
dl+d2+
...+
d,=d.
Gi en a posi i e in ege
x,
le
diJ
deno e
he
i h
bi o he bina y ep esen a ion o
x.
The
leas
signi ican bi is conside ed o
be
he
0 h
bi . We
also
de ine Kjin
he
ollowing way.
K1=0,
and
o
e e y
I<jG+l
we ha e ha :
j-
1
Kj
=
di
i=
1
Session
2:
E icien Design Me hods
I
0
1
2
3
4
5
6
7
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15
8
9
10
11
11111111111
-
23
Figu e
5:
A
xo
embedding
o
a
4-cube
on o a ing.
The
labels
indica e which
node
o
he dcube
is
mapped
on o he co esponding node
o
he ing.
Le
G
be
he
g aph which ep esen s
he
d-cube and
T
be he g aph which ep esen s he o us.
men, e ex
n
o
G
is mapped on o e ex
(mpa
...,mc)= m
in)
in
T
as
ollows:
mJ{i)
=
n(i+Kj)
mJ{dj
-
2)
=
XOR
(n(Kj+l-
I),
n(Kj+,
-
2))
i
E
10,
dj
-
I],
i
dj
-
2
Fo
he
pa icula
case
o a (2dc,
...,
99
c-dimensional o us,
he
dis ances co esponding o
his
embedding a e:
Di
=
zd/c-2
D.
=
$m~ddc
o he wise
i
i
=
1
(ac)
-
I,
0
e
1
5
c
and he a e age dis ance
is:
d/c-2
.
2dIC-2
[
-k
=O
2’)
- -
c(2d/c-2+2d/c-1-
1)
d d
D,
=
Figu e
6
shows an example o
d=6.
No e
he
simplici y o unc ion ’Jn).
?his
unc ion, which
is used e y equen ly o ou ing messages du ing he execu ion o
he
CC
d-cube algo i hm,
consis s
o simple bi ope a ions
and
i s compu a ional cos is negligible.
5.
P oo
o
op imali y
o ,,,
o
ings
Ou
c i e ion
o
measu e he goodness o any embedding wi h cons an dis ances is i s a e age
dis ance
as
de ined in sec ion 2, since minimizing he a e age dis ance implies minimizing he
execu ion ime
o
CC
d-cube algo i hms. In
his
sec ion we p o e ha he xo embedding has he
minimum a e age dis ance o emwngs wi h cons an dis ances o hype cubes
on o
ings.
To
show ha he xo embedding is op imal o ings, we will p o e ha he a e age dis ance
o
any embedding wi h cons an dis ances is highe han o
equal
o
he a e age dis ance o heh,
embedding.
This
is
s a ed
by heo em
10.
Be o e
his
heo em we p esen se e al lemmas and
co olla ies
ha
a e needed
o
p o e
ha
esul Fi s we ind a lowe bound o he sum o any
se
o
d-ldis ances
co esponding
o
any
embedding wi h cons an dis ances. men, we
ind
a lowe
bound o he highes dis ance o he embedding. Bo h oge he gi e a lowe bound o he a e age
dis ance o any embedding wi h cons an
dis ances.
This
lowe bound is
he
a e age dis ance o he
m
embedding, which p o es i s op imali y.
De ini ion:
Gi en any node o a hype cube, we de ine
Ndn).
whe e
D
is any subse o
dimensions o he hype cube,
as
he
node
ha we each i we
s a
a node
n
and we mo e h ough
e e y dimension in
D,
one a e
ano he ,
using each dimension exac ly once
(as
we know, he o de
in which
he
dimensions
a e
used
does
no ma e ,
he
esul will be
he
same).
Fo
ins ance,
i
D=[l,3),
hen
Ndn)
=
N3(Nl(n))
=
NI(N3(n)).