Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
4, N
o
. 4
,
A
ugu
st
2014
, pp
. 63
1
~
63
6
I
S
SN
: 208
8-8
7
0
8
6
31
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Ranking in Distributed Unce
rtain Database Environments
Yousry M.
Abdul Az
eem,
Al
i I.
ElDes
o
uky,
and Hesham A. Ali
Com
puters
Engi
neering
and
S
y
s
t
em
s
Departm
e
nt
, F
acu
lty
of
Engineering
,
Mansou
ra University
, Eg
y
p
t
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ma
r 4, 2014
Rev
i
sed
Jun
26,
201
4
Accepte
d
J
u
l 10, 2014
Distributed
d
a
ta processing
is a major
field in
n
o
waday
s
applications. Man
y
applications collect
and
process data from distr
i
but
ed
nodes to gain
over
a
ll
results. Larg
e amount of data transf
er and network delay
made data
processing in a centr
alized
manner a hard operation repr
esenting an
important problem. A ver
y
co
mmon
way
to solve this problem is ranking
queries. Rank
in
g or top-
k
qu
eries concentr
ate
only
on
the h
i
g
h
est rank
ed
tuples a
ccord
in
g to user’s in
tere
st. Another
issue in most nowaday
s
appli
cat
ions
is
data unc
ert
a
int
y
. M
a
n
y
techn
i
q
u
es
were intro
duced for
modeling, managing, and
processing uncertain
datab
a
ses. Alth
ough these
techn
i
ques
were
effi
cien
t,
the
y
d
i
dn’t de
al with
distributed
da
ta
unc
e
r
ta
inty
.
This paper d
e
als with both data u
n
certa
inty
and distribution bas
e
d
on ranking
queries. A novel framework is
proposed
for ranking distribu
ted uncertain
data. The framework has a suite of novel algor
ithms for ranking data and
monitoring
updates. These algor
ithms
help in
reducing th
e
communicatio
n
rounds used and am
ount of dat
a
transm
itted w
h
ile achi
eving effici
ent and
effective rankin
g
. Experime
ntal results show th
at th
e proposed
framework
has
a gr
eat
im
pact
in r
e
ducin
g com
m
unicatio
n cos
t
com
p
are
d
to oth
e
r
techn
i
ques
.
Keyword:
Datab
a
se app
licatio
n
s
Di
st
ri
b
u
t
e
d
da
t
a
bases
R
a
nki
ng
Top-
k
qu
er
y
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Yous
ry
M. Abdul Azeem
,
C
o
m
put
ers E
n
gi
nee
r
i
n
g a
n
d
Sy
st
em
s Depa
r
t
m
e
nt
, Facul
t
y
of
En
gi
nee
r
i
n
g
,
M
a
ns
o
u
ra
U
n
i
v
ersi
t
y
D
a
q
a
h
lia 355
16
,
Eg
y
p
t
Em
a
il: yo
u
s
ry@m
an
s.ed
u.eg
1.
INTRODUCTION
Di
st
ri
b
u
t
e
d
da
t
a
pr
ocessi
n
g
i
s
a m
a
jor fi
e
l
d i
n
no
wa
da
y
s
appl
i
cat
i
o
n
s
. M
o
st
of
t
h
e com
m
on
appl
i
cat
i
o
ns col
l
ect
and pr
o
cess dat
a
fr
om
di
st
ri
but
e
d
si
t
e
s t
o
gai
n
an
ove
ral
l
resul
t
.
Exam
pl
es of suc
h
appl
i
cat
i
o
ns ar
e;
C
ont
e
n
t
Di
s
t
ri
but
i
o
n
Net
w
or
k
(C
D
N
)
[
1
,
2]
, se
ns
or
ne
t
w
o
r
ks
[
3
,
4,
5,
6]
, m
u
l
t
i
m
e
di
a
dat
a
base
[
7
]
,
i
n
f
o
rm
at
i
on re
t
r
i
e
val
fr
om
geo
g
r
ap
hi
cal
l
y
separat
e
d
da
t
a
cent
e
rs [
8
,
9,
10]
,
net
w
o
r
k
m
o
n
ito
r
i
n
g
over
d
i
str
i
bu
ted lo
g
s
[11
]
and data ex
tr
acted
fr
o
m
a set of
data str
eam
s [
1
2
,
13
]. Cen
t
r
a
l
i
zed
pr
ocessi
ng
of
si
t
e
s’ dat
a
i
s
a very
ex
pe
nsi
v
e t
a
sk. M
o
re
o
v
er
, l
a
rge am
ount
of
dat
a
t
r
a
n
sfe
r
a
nd
net
w
o
r
k
d
e
lay m
a
k
e
s it
h
a
rd
to
m
a
n
i
pu
late th
ese ap
plicatio
n
s
in
a cen
t
ralized
m
a
n
n
e
r
[1
].
Usu
a
lly, in
m
o
st o
f
th
ese
applications, t
h
e user
does not nee
d
to
process all da
ta in
th
e system
. In
stead
,
o
n
l
y a fraction
o
f
d
a
ta is
retu
rn
ed
to
th
e u
s
er acco
r
d
i
ng
to
h
i
s i
n
terest. Th
is is
o
f
ten d
o
n
e
b
y
u
s
ing r
a
nk
ing
qu
er
ies. Rank
ing
or
to
p-
k
que
ri
es ret
u
rn
onl
y
t
h
e hi
g
h
est
ran
k
e
d
k
tuples according to a use
r
-def
i
n
ed sc
ori
n
g function. M
a
ny
t
echni
q
u
es we
r
e
desi
gne
d t
o
r
a
nk
di
st
ri
b
u
t
e
d
dat
a
i
n
t
r
od
uci
ng m
a
ny
effi
ci
ent
al
gori
t
hm
s
[3
, 7, 8
,
9, 1
1
,
12]
.
Ho
we
ver
,
n
o
n
e
of t
h
ese t
e
c
h
ni
q
u
es ha
s dea
l
t
wi
t
h
di
st
rib
u
ted
un
certain
data. Ev
en
t
h
oug
h, in
m
o
st o
f
th
e
n
o
w
a
d
a
ys
ap
p
l
icatio
n
s
, d
a
ta
i
s
fu
zzy
or
un
cer
t
ain
[14
,
15
, 16
, 17
, 18
, 19
, 20
, 21
].
Top-
k
qu
eries
in
un
certai
n
datab
a
se system
aim
at
fi
ndi
n
g
t
h
e
hi
ghest
ran
k
e
d
k
tuple
s
over al
l
p
o
s
sib
l
e in
- stan
ces of th
e d
a
t
a
b
a
se. Th
ese i
n
stan
ces
are called
p
o
ssi
b
l
e wo
rl
d
s
. Pro
cessin
g
th
is
q
u
e
ry will
consum
e both tim
e
and communication bandwidth. Ma
ny recent a
p
proac
h
es ha
ve
dealt with ra
nki
ng
cen
tr
alized
un
cer
t
ain
datab
a
se [
2
2
,
23
,
24
,
25]. H
o
w
e
v
e
r,
there are on
ly two
ap
pro
a
ch
es th
at h
a
v
e
d
ealt
with
ran
k
i
n
g
i
n
di
st
ri
b
u
t
e
d u
n
cert
a
i
n
sy
st
em
s [1
5,
2
6
]
.
T
h
ese
were
the
first
approaches
to
address
ra
nki
ng in
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
63
1
–
63
6
63
2
unce
r
t
a
i
n
di
st
r
i
but
ed e
n
vi
ro
n
m
ent
,
but
t
h
ey
had s
o
m
e
drawbac
k
s
.
In t
h
e
fi
rst
app
r
oach
t
h
e
m
a
i
n
con
cern
was th
e transmissio
n
b
a
ndwid
th
trad
ing
-
o
f
f laten
c
y.
The central server c
o
mm
uni
c
a
tes with
o
t
h
e
r no
des
several tim
es
per
one tuple e
ach access,
for com
puting top-
k
list. This l
a
rge
num
ber of accesses inc
r
eases
t
h
e l
a
t
e
ncy
.
T
h
e sec
o
n
d
a
p
p
r
oac
h
was a
p
p
l
i
e
d m
a
i
n
l
y
on
wi
rel
e
ss se
ns
or
net
w
o
r
ks
t
h
at
are di
vi
de
d
i
n
t
o
cl
ust
e
rs.
Eac
h
cl
ust
e
r
hea
d
se
nds
pre
-
p
r
u
n
e
d
set
o
f
t
upl
es
(
s
uf
fi
ci
ent
set
)
t
o
t
h
e
que
ry
no
de.
Each
s
u
f
f
i
c
i
e
nt
set needs a lot
of c
o
m
putation whic
h is
done
at each
cluste
r head.
More
ove
r, bot
h
a
p
proac
h
es don’t
m
onitor
u
p
d
a
tes i
n
tup
l
e v
a
lu
es or
rank
s.
Th
is
p
a
p
e
r
p
r
op
o
s
es a
n
e
w fram
ewo
r
k
t
h
at will h
e
l
p
i
n
o
v
e
rco
m
in
g
th
ese
d
r
awb
ack
s. The fram
e
wo
rk
utilizes o
n
l
y o
n
e o
r
t
w
o
roun
ds o
f
co
mm
u
n
i
catio
n
in
t
h
e pro
p
o
s
ed
algo
rit
h
m
s
.
So, t
h
e am
ou
nt
of t
r
a
n
sm
i
tted dat
a
i
s
m
i
ni
m
i
zed fo
r achi
e
vi
n
g
ef
fi
ci
ent
di
st
ri
b
u
t
e
d
unce
r
t
a
i
n
dat
a
bas
e
ran
k
in
g.
The rest
o
f
t
h
i
s
paper i
s
or
gani
ze
d as fol
l
ows. Sect
i
o
n
2 pre
s
ent
s
t
h
e
pro
p
o
se
d fra
m
e
wor
k
an
d
defi
nes t
h
e u
s
e
d
dat
a
m
odel
.
Sect
i
on
3 di
sc
usses t
h
e
det
a
i
l
s
of t
h
e
pr
o
p
o
s
ed al
g
o
ri
t
h
m
s
. Sect
i
on 4
val
i
d
at
es
the proposed a
l
gorithm
s
and evalua
tes thei
r efficiency and pe
rform
a
nc
e
by
a com
p
reh
e
nsi
v
e e
x
peri
m
e
nt
al
st
udy
.
Fi
nal
l
y
,
t
h
e pa
pe
r i
s
c
o
ncl
u
ded
i
n
Sec
t
i
on
7.
2.
THE PROPOSED FRAME
W
ORK
The c
r
uci
a
l
p
r
obl
em
s faci
n
g
t
h
e c
u
r
r
ent
a
p
pr
oac
h
es
of
di
st
ri
but
e
d
u
n
cer
t
a
i
n
dat
a
base
r
a
nki
ng
are
co
nsid
ered
as
fo
llows: (
i
)
Th
e num
ber
of t
upl
es eac
h si
t
e
cho
o
ses t
o
se
nd a
s
a local a
n
swe
r
to t
h
e
query.
Large
num
ber of t
u
pl
es co
ns
um
es bot
h net
w
o
r
k
ban
d
w
i
d
t
h
an
d t
i
m
e
; al
so som
e
t
upl
es m
a
y
not
be i
n
v
o
l
v
e
d
i
n
t
h
e ran
k
i
n
g pr
ocess
.
On t
h
e ot
her ha
n
d
, s
m
al
l num
ber of t
upl
es m
a
y cause i
n
co
nsi
s
t
e
n
t
ranki
n
g
res
u
l
t
s
. (
ii
)
Th
e nu
m
b
er of co
mm
u
n
i
cati
o
n
rou
n
d
s
(phases) app
lied
in
rank
ing
pro
c
ess will affect also
th
e co
n
s
u
m
e
d
n
e
two
r
k
b
a
ndwid
th. So
, it is requ
ired
to
m
i
n
i
mize th
e n
u
m
b
er o
f
co
mmu
n
i
cation
ph
ases. (
iii
) M
o
re
o
v
er
, The
n
eed
fo
r m
o
n
ito
ri
n
g
p
r
o
cess i
s
also
an
im
p
o
rtan
t issu
e. Only effectiv
e upd
ates in
v
a
lu
es an
d
prob
ab
ilit
ies o
f
t
upl
es at
di
f
f
er
ent
si
t
e
s s
h
o
u
l
d
be
rep
o
rt
e
d
t
o
que
ry
no
de t
o
up
dat
e
t
h
e
c
u
rre
nt
q
u
e
r
y
ans
w
er
.
The
pr
op
ose
d
fram
e
wor
k
ai
m
s
t
o
sol
v
e t
h
e pre
v
i
o
usl
y
d
i
scusse
d p
r
o
b
l
e
m
s
. The m
a
in feat
ures
of
this fram
e
- wo
rk a
r
e: (
i
) Fi
xi
ng t
h
e n
u
m
b
er of c
o
m
m
uni
cat
i
on r
o
u
n
d
s (
o
nl
y
one
ro
u
n
d
)
. (
ii
) M
i
ni
m
i
zing t
h
e
num
ber of candidate tuples.
Candi
date
tupl
es are the set of tuples each
node will send to query
node for
ran
k
i
n
g pr
oces
s.
(
iii
) M
o
ni
t
o
r
i
ng t
u
pl
e
up
dat
e
s i
s
a m
a
i
n
ph
ase aft
e
r c
o
m
put
i
ng t
h
e t
o
p-
k
list. Th
e fram
e
w
ork
co
nsists o
f
three
m
a
in
layers,
Qu
ery
,
R
ank
i
n
g
, a
n
d
M onito
rin
g
layers. In
th
ese layers a su
ite of no
v
e
l
alg
o
rith
m
s
are in
tr
odu
ced and app
lied
.
The
Query
layer is a starti
n
g
po
in
t fo
r t
h
e fram
ewo
r
k in
wh
ich
th
e d
i
stribu
ted
top
-
k
que
ry
i
s
form
ulated then broa
dcaste
d to all nodes
.
T
h
is quer
y m
a
y be accom
p
anied with a thres
hol
d val
u
e. Process
execution i
n
t
h
is layer takes
place at the side
of the
Query
n
ode
.
In the
Ran
k
ing
layer each site, on
receivi
ng t
h
e query, starts
com
puting the
neede
d
tuples for query
an
swer
.
I
f
th
e
q
u
e
r
y
h
a
s a thresho
l
d
v
a
l
u
e th
en
a
p
r
un
ing
p
h
a
se is co
nducted
f
i
r
s
t,
o
t
h
e
r
w
ise co
n
tinu
e
w
ith
t
h
e ra
nki
ng
pr
o
cess. T
h
e m
a
i
n
phas
e
i
n
t
h
i
s
l
a
y
e
r i
s
t
h
e ans
w
er c
o
m
put
at
i
on
p
h
ase w
h
i
c
h i
s
m
a
naged e
n
t
i
r
el
y
usi
n
g t
h
e
ne
wl
y
pr
op
ose
d
al
g
o
ri
t
h
m
Thres
h
ol
d
-
t
u
ned
Di
st
r
i
but
ed
T
o
p
-
k
(TD
T
k
).
Afte
r c
o
m
puting t
h
e
needed
tuples for que
r
y answer, each site
sends its local answe
r
to the
Query
node to
co
m
p
u
t
e t
h
e d
i
str
i
bu
ted
t
o
p-
k
list (DT
k
).
The
kth
tup
l
e in DT
k
is
b
r
o
a
dcasted
to
all t
h
e
n
o
d
e
s t
o
st
art th
e m
o
n
itoring
ph
ase. Pro
cess
ex
ecu
tion
i
n
t
h
is layer is don
e at bo
th
sid
e
s
of th
e
Query
n
o
d
e a
n
d
ot
her
n
ode
s si
m
u
l
t
a
neousl
y
.
In t
h
e
Mon
ito
ri
n
g
layer, eac
h
node as
signs its lowe
r
bound
t
uple
with the
received
one.
T
h
is tuple is
th
e corn
er st
o
n
e in
th
e m
o
n
itoring
p
r
ocess.
T
h
e t
u
pl
es are
r
a
nke
d at
eac
h
si
t
e
consi
d
eri
n
g l
o
wer
b
o
u
n
d
t
upl
e
in
th
e rank
ing pro
cess.
Tup
l
es ran
k
ed lower th
an
l
o
w
e
r
bo
u
n
d
t
u
pl
e a
r
e assi
g
n
e
d
as
candi
dat
e
l
i
s
t
.
Any
change i
n
ca
ndidate list val
u
es
or any
ne
w
values
upd
ated
with a
rank
lower th
an
l
o
wer
bo
und
tup
l
e is
co
nsid
ered a can
d
i
d
a
te to b
e
in
top
-
k
. T
h
i
s
t
upl
e i
s
se
nt
t
o
que
ry
no
de t
o
up
dat
e
t
h
e c
u
r
r
e
nt
D
T
k
.
Moni
toring
pha
se i
s
a c
o
nt
i
n
u
o
u
s
pr
ocess
execut
e
d
on
n
o
d
es
wi
t
h
t
upl
es
’
up
dat
e
s a
n
d
Quer
y
no
de.
2.
1. D
a
t
a
M
o
d
e
l
Defi
ni
ti
on
Co
n
s
i
d
er
an
un
certain
d
a
tabase
D
,
ho
r
i
zontally d
i
str
i
b
u
t
ed
ov
er
a set of n
o
d
e
s
M
with size |M| =
m
suc
h
t
h
at
⋃
. A central server
R
works as
t
h
e
Qu
ery
no
de.
T
h
e
dat
a
base
co
nsi
s
t
s
o
f
N
t
u
pl
es
fra
gm
ented with di
ffe
rent size
s suc
h
that f
o
r
any
site
M
i
, |
D
i
| =
N
i
and
∑
. Tuples in
D
i
are
denote
d
as
⋃
and t
h
ei
r
score
val
u
es a
s
ran
d
o
m
vari
abl
e
s
⋃
a rand
om
vari
abl
e
X
ij
’s
pdf
is
represen
ted
wi
th
th
e p
a
irs (
v
jx
,
p
jx
) where 1
x
B
i
and
B
i
is the bounde
d
size of
X
ij
. T
h
e m
a
in object
ive of
th
is work
is to repo
rt at
R
the answer
of a
Distributed top-
k
qu
ery.
Formally, Defin
ition
1
Distribu
ted to
p-k
que
ry
(
DT
k
)
:
I
t
i
s
t
h
e
q
u
ery
t
h
at
ret
u
r
n
s t
h
e
t
o
p
-
k l
i
s
t
o
f
t
upl
es
wi
t
h
t
h
e
hi
g
h
est
ran
k
a
m
ong al
l
di
st
ri
but
e
d
no
des
.
Let
D
b
e
an u
n
cert
a
i
n
dat
a
base
di
st
ri
but
e
d
ove
r
M
no
des suc
h
t
h
a
t
⋃
, let
f
b
e
th
e
r
a
nk
ing
fu
nct
i
o
n.
DT
k
que
ry
ret
u
r
n
s t
h
e m
i
n k t
u
pl
es
suc
h
t
h
at
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ran
k
ing
in Distrib
u
t
ed Un
certa
in
Da
t
a
ba
se
En
vironm
en
ts (You
sry M. Abdu
l Azeem
)
63
3
min
…
∈
∀
,
≺
(1
)
Whe
r
e
ti
≺
f tj
m
eans that
ti
dom
inates
tj
according t
o
ra
nki
ng function
f
. If
t
h
e r
a
nk
i
n
g fun
c
tio
n used
is
Expected rank then
the
DT
k
-
q
uery
ret
u
r
n
s t
h
e top
-
k
l
i
s
t
wi
t
h
t
h
e l
o
west
e
x
pect
ed
ran
k
a
m
ong al
l
di
st
ri
but
e
d
no
des
M
.
3.
THE PROPOSED
ALGORITHMS
In
th
is section
,
we
d
i
scuss th
e p
r
o
p
o
s
ed su
ite o
f
algorith
m
s
wh
ich
are i
m
p
l
e
m
en
ted
in
ou
r
f
r
a
m
e
w
o
rk
. Th
e two pr
opo
sed
algor
ithm
s
are em
ployed
in the
Ra
nki
n
g
layer
.
Each
o
n
e
of
t
h
e
r
a
nk
ing
al
go
ri
t
h
m
s
i
s
im
pl
em
ent
e
d t
o
ran
k
di
st
ri
but
e
d
dat
a
base
at attrib
u
t
e-lev
e
l un
certain
ty. Th
e first alg
o
rith
m was
in
trodu
ced in
a prio
r work [2
7].
3.
1. Th
resh
o
ld
-tu
ned
Distrib
u
ted T
o
p
-
k
al
gori
t
hm (T
DT
k
)
In t
h
is algorithm
,
a user-defined
threshol
d value is
used to pr
une tupl
es at each node. The
rest
tuples a
r
e
orde
red accordi
n
g t
o
th
ei
r e
xpecte
d
rank t
h
en the top-
k
list,
from
each node, is sent t
o
t
h
e
query
no
de.
The
q
u
er
y
no
de ra
n
k
s t
h
e t
u
pl
es t
o
get
DT
k
th
en
b
r
o
a
d
cast
th
e
kt
h
tup
l
e in
DT
k
to
all n
o
d
e
s. Th
is
v
a
lu
e
i
s
used t
o
m
odi
fy
l
o
cal
l
o
wer
bo
un
ds use
d
f
o
r m
oni
t
o
ri
n
g
pha
se. Fo
rm
al
ly
, a DT
k
query
i
n
TDT
k
alg
o
rith
m
Qk
retu
rn
s th
e
h
i
gh
est rank
ed
k
tu
p
l
es su
ch
t
h
at:
∈
|
(2
)
Whe
r
e:
⋃
,
A
x
i
s
t
h
e set
of
hi
ghe
st
score
k
-
t
upl
es f
r
om
si
t
e
x
or :
⋃
∈
|
and
r
(
t
i
)
as
i
n
equat
i
o
n 2 wi
t
h
p
ij
(t
he t
h
resh
ol
d
val
u
e)
.
The
det
a
i
l
e
d
s
t
eps
of
TDT
k
algorithm
ar
e
as fo
llows: The
Query
nod
e
R
br
oadcast
s
a
t
op-
k
qu
er
y
Q
k
wi
t
h
pr
uni
ng
t
h
resh
ol
d
val
u
e
, to
all n
o
d
e
s. It
in
itializes th
e p
r
io
rity qu
eu
e
DT
k
and t
upl
e
LT
with em
pty values. Eac
h
node
M
i
M
emp
ties a lo
cal p
r
io
rity
que
ue
A
i
and
in
itializes lo
wer bou
nd
LB
T
i
with
em
p
t
y v
a
lu
e. Tup
l
es’
v
a
lu
es
with
pro
b
a
b
ilities less th
an
are
om
it
t
e
d. The t
u
pl
es are t
h
en
ra
nke
d acc
or
di
n
g
to their e
x
pected ra
nks a
nd i
n
serted i
n
to
A
i
.
The first
k
tup
l
es in
A
i
are sent to
R
an
d
in
serted in
DT
k
.
LBT
i
is set with
th
e
k
th
t
upl
e i
n
A
i
. Aft
e
r
c
o
m
put
i
ng DT
k
,
R
broad
casts
th
e last tup
l
e
LT
t
o
al
l
ot
he
r
n
ode
s. Eac
h
n
o
d
e
, set
s
LB
T
i
with the
recei
ved
LT
an
d starts
m
o
n
ito
rin
g
ph
ase.
3
.
2
.
Ceiling-tuned Distribute
d To
p-k Al
g
o
r
ithm
(CDTk)
In t
h
is algorithm
,
each node
m
a
intain
s a pri
o
rity queue
with local top-
k
li
st ranked
by expected
rank.
The first
tuple
s
from
each node is sent to
query node.
The
rest tuples are c
a
lled candi
date
list used later to
refine
D
T
k
.
Th
e q
u
ery
n
ode
r
a
nk
s t
h
e
t
u
pl
es
t
o
get
fi
rst
ph
ase DT
k
. T
h
e
k
th
ran
k
e
d
t
upl
e i
n
DT
k
is sent to
all
no
des
o
f
t
h
e
fi
rst
(
k
-
1
)-t
upl
es in
DT
k
. Eac
h
node
ra
nks t
h
e t
uples
in ca
ndi
date list conside
r
ing t
h
e
receive
d
t
upl
e fr
om
Qu
ery
no
de, t
h
e
n
sen
d
s al
l
t
upl
es ran
k
ed l
o
we
r
t
h
an t
h
at
t
upl
e
.
Aft
e
r
w
a
r
ds
, Que
r
y
n
ode r
a
nks t
h
e
tu
p
l
e fo
r th
e seco
nd
ti
m
e
to
g
e
t th
e actu
a
l DT
k
. Ag
ain
,
th
e
last tu
p
l
e in
DT
k
is used,
but
to specify local
lower
b
oun
d in
ord
e
r
to
start m
o
n
itoring
p
h
a
se.
A DT
k
qu
er
y in
CD
T
k
al
go
ri
t
h
m
i
s
form
ul
at
ed base
d o
n
e
quat
i
o
n
3. It
re
t
u
r
n
s t
h
e
hi
g
h
e
s
t
ran
k
ed
k
tuples with
t
h
e lowest
e
x
pected rank
am
ong
all sites, in two succes
sive
phases. Form
ally,
∈
,
(3
)
whe
r
e
A
is th
e set o
f
tup
l
es with
th
e lowest ex
p
ected
rank
fro
m
all n
o
d
es,
,
A
x
is th
e
set o
f
tuples
with lowest expecte
d
rank
fr
om
no
de
x s
u
ch t
h
at
:
i
n
t
h
e
1
st
pha
se an
d
A
x
=
i
n
t
h
e
2
nd
pha
se, whe
r
e
, r
(
t
i
) is expect
ed ra
nk
of
t
i
as
in
equat
i
o
n 4
a
n
d
LT
is the
k
th
ra
nke
d t
upl
e i
n
t
h
e
fi
rst
phase
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
63
1
–
63
6
63
4
The det
a
i
l
e
d s
t
eps of C
D
Tk
al
go
r
ithm
are
as follows: The Query node R co
m
putes the num
ber of t
uples
neede
d
from
each site C. Afterwa
r
ds
it broadcasts a
top-k query
(specifying
C) to all sites. It initializes the
p
r
i
o
rity q
u
e
u
e
DTk
and
tu
p
l
e LT with
e
m
p
t
y v
a
lu
es. Each n
o
d
e
Mi 2
M
in
itializes
th
e p
r
i
o
rity q
u
e
u
e
A
i
and
t
upl
e
LB
T
i
wi
t
h
em
pt
y
val
u
es. It
ra
nk
s i
t
s
ow
n t
u
pl
es acc
or
di
n
g
t
o
t
h
ei
r
expect
e
d
ra
n
k
and se
n
d
s t
h
e
fi
rst
C
t
upl
es t
o
R
wh
ile k
eep
ing
the rest tu
p
l
es i
n
A
i
(can
didate list) f
o
r
fu
rthe
r D
T
k
refi
nin
g
.
At
R
, tuples are ranke
d
by
expect
e
d
ra
nk t
h
e
n
t
h
e
k
th
ran
k
e
d
tup
l
e is sen
t
to
all
th
e sites o
f
th
e first (
k
-1
)-t
uple
s
in DT
k
. Eac
h
n
ode
assigns
LBT
i
with
LT
th
en
rank
s
A
i
with
LB
T
i
. Tu
ples ran
k
e
d
lowe
r tha
n
LB
T
i
are sent to
R
. A
not
her ra
n
k
i
n
g
pha
se i
s
do
ne t
o
get
t
h
e re
fi
ne
d
DT
k
. The
ne
w val
u
e o
f
LT
is com
puted and broa
dcaste
d to all the nodes. Each
no
de set
s
LB
T
i
with
LT
value and
starts
m
oni
toring phase.
4.
EX
PER
I
M
E
NTA
L
STUDY
A d
a
ta g
e
n
e
rato
r is d
e
v
e
lop
e
d
in
order to
gen
e
rate syn
t
h
e
tic d
a
tasets u
s
ed
in
algo
rith
m
s
v
e
rifying
.
It
gene
rates sy
nthetic Ga
ussian dataset
whe
r
e
each
rec
o
rd’s
score attribut
e draws
its val
u
es
from
a Ga
ussian
distribution. For
each rec
o
rd
, the sta
nda
rd deviation
is
random
ly selected
from
[1,1000] and t
h
e m
e
a
n
is
ran
d
o
m
l
y
selected fr
om
[5
,1
000
00
]. Each r
eco
rd
h
a
s
g
choi
ces
fo
r i
t
s
score
val
u
es
whe
r
e g i
s
ra
n
dom
ly
selected
fro
m
1
to 5. Th
e
g
e
nerato
r con
t
ro
ls
b
o
t
h
sco
r
e v
a
l
u
es and
p
r
ob
ab
il
ities o
f
th
e
g
e
nerated
t
u
p
l
es.
4.
1. E
ffec
t
of
Di
ffere
nt
Si
z
e
s T
o
p
-
k
Lists
k
Fi
gu
re 1 s
h
ow
s t
h
e com
m
uni
cat
i
on cost
s
of
t
h
e bot
h p
r
o
p
o
se
d al
go
ri
t
h
m
s
fo
r di
f
f
ere
n
t
val
u
es
of
m
(1
0, 5
0
,
10
0,
20
0,
50
0 an
d 10
0
0
n
odes
)
.
Each fi
g
u
r
e co
m
p
ares bet
w
een t
h
e b
o
t
h
al
g
o
ri
t
h
m
s
repres
ent
i
n
g
TDT
k
wi
t
h
ze
r
o
t
h
res
hol
d val
u
e (
=
0)
.
I
t
is ob
serv
ed
fr
om
f
i
g
u
r
e 1 th
at th
e co
mm
u
n
i
catio
n
co
st of
TD
T
k
increases linea
rly with
k
unt
i
l
k
=
N/m
(num
ber of tuples
at each site).
After that val
u
e the comm
unication
cost is fi
xe
d for any
greater
k
. Each
site in
this case will send
all its
tup
l
es
so
th
e co
mm
u
n
i
catio
n
co
st
will b
e
the sam
e
. The
comm
unication c
o
st of C
D
T
k
also
in
creases lin
early with k
bu
t with
a lo
wer
rate so th
at it
appea
r
s, c
o
m
p
ared to
TDT
k
,
as i
f
i
t
has a
fi
xed
val
u
e.
Fi
g
u
re
1
al
so
sh
o
w
s t
h
at
, t
h
e
c
o
m
m
uni
cat
i
on c
o
st
of
A-
BF
i
s
fi
xed
f
o
r
any
val
u
e
of
k
.
4.
2.
E
ffec
t
of
Di
ffere
nt Number
of T
uple
s
N
The
e
ffect o
f
diffe
re
nt
N
(t
o
t
al
num
ber
of
t
upl
es i
n
t
h
e
dat
a
base
)
on
t
h
e c
o
m
m
uni
cat
i
on c
o
st
i
s
st
udi
e
d
he
re fo
r a fi
xe
d n
u
m
b
er of si
t
e
s o
n
whi
c
h t
upl
es a
r
e di
st
ri
b
u
t
e
d
.
Fi
gu
re 2a s
h
o
w
s t
h
e com
m
uni
cat
i
o
n
co
st of
t
h
e two p
r
op
osed algor
ith
m
s
w
ith
m
= 1
0
a
nd
k
=
10;
k
=
1
00 a
n
d
di
ffe
re
nt
val
u
e
s
o
f
N
. T
h
e
observe
d
com
m
uni
cat
i
on c
o
st
of
T
D
T
k
in
creases linearly with
N
un
til
N
=
1
000
th
en it g
e
ts stab
le (sam
e v
a
lu
e wit
h
mo
r
e
N
),
whi
l
e
t
h
e c
o
m
m
uni
cat
i
on c
o
st
of C
D
T
k
i
n
creases
linearly with
N
.
The
reaso
n
o
f
t
h
i
s
be
ha
vi
o
r
i
n
TDT
k
i
s
t
h
at
e
ach si
t
e
se
nds
k t
u
pl
es t
o
be
pr
ocesse
d.
At
l
o
we
r
val
u
es
of
N
(when num
b
er of tupl
es at each site is less tha
n
k
) each
site send
s all its tu
p
l
es, so
that th
e
co
mm
u
n
i
catio
n
co
st in
creases u
n
til
N
=
m
x
k
. Inc
r
easi
ng
N
mo
r
e
t
h
a
n
m
x
k
will no
t affect nu
m
b
er o
f
tu
p
l
es
sent
w
h
i
c
h i
s
k
. O
n
t
h
e ot
her
han
d
C
D
T
k
shows a linea
r increase with t
h
e
in
crease of N
although
the num
b
er
of t
u
pl
es sent
i
s
al
way
s
t
h
e sam
e
(
k/
m
).
Fi
gu
re 2
b
sh
o
w
s t
h
e com
m
uni
cat
i
on c
o
st
o
f
t
h
e t
w
o
pr
o
p
o
se
d
alg
o
rith
m
s
b
u
t
with
m
= 10
0 and
k
=
10;
k
= 10
0. TD
T
k
h
a
s t
h
e sam
e
behavi
or a
s
wi
t
h
m
= 1
0
, wh
ile CDT
k
shows a ve
ry low c
o
mm
unication cost com
p
are
d
to T
D
T
k
. The di
st
ri
but
i
on o
f
t
h
e sam
e
num
ber o
f
t
upl
es
am
ong la
rge
r
num
ber
of site
s increases t
h
e
probability of
getting t
o
p-k
quic
k
ly or
at l
east havi
ng a
higher
lo
wer bo
und
in th
e b
e
g
i
nn
ing
o
f
th
e secon
d
ph
ase of th
e
al
g
o
ri
t
h
m
.
Fi
gu
re
2a an
d
2b s
h
ow also a com
p
arison
bet
w
ee
n T
D
T
k
, CDT
k
and
A-BF
to observe
the
effect of di
ffe
rent
N
on e
ach algorithm
.
As
seen from
both
fi
g
u
res
t
h
e c
o
m
m
uni
cat
i
on c
o
st
of
A-BF
in
creases lin
early
with
a
v
e
ry h
i
gh
rate.
4.
3.
E
ffec
t
of
Di
ffere
nt
Number of Distri
buted
Sites
m
The effect of
m
o
n
t
h
e c
o
m
m
uni
cat
i
on co
st
i
s
st
udi
ed
he
re f
o
r a
fi
xe
d
num
ber
of t
u
pl
es di
st
ri
b
u
t
e
d
on
di
ffe
rent
si
t
e
s. Fi
gu
re
2c s
h
o
w
s
t
h
e c
o
m
m
uni
cat
i
on cos
t
of
t
h
e t
w
o
pr
op
ose
d
al
go
ri
t
h
m
s
wi
t
h
N
=
100000
and
k
= 10;
k
= 10
0 an
d di
ff
erent
val
u
es of
m
. The obse
rved comm
unication cost of TDT
k
increa
ses linearl
y
with the i
n
crea
se of
m
,
wh
ile
th
e co
mm
u
n
i
catio
n
co
st of C
D
T
k
decreas
es also
linearly
with
th
e in
crease o
f
m
.
The m
a
t
c
hi
ng
poi
nt
o
f
t
h
e t
w
o al
g
o
ri
t
h
m
s
i
s
at
m
= 15
0
or
m
= 75.
At the
s
e values
, the
comm
unication costs
of
bot
h algorit
h
m
s
are nearly equal.
Afte
r these values
, TDT
k
ten
d
s
t
o
in
crease wh
ile CDT
k
de
creases
. Figure
2c s
h
ows that,
the comm
unication c
o
st of
A-
B
F
inc
r
eases l
i
nearly but wit
h
hi
ghe
r val
u
es
t
h
an b
o
t
h
T
D
T
k
and
CDT
k
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ran
k
ing
in Distrib
u
t
ed Un
certa
in
Da
t
a
ba
se
En
vironm
en
ts (You
sry M. Abdu
l Azeem
)
63
5
Fig
u
r
e
1
.
Co
mm
u
n
i
catio
n
co
sts o
f
th
e
pr
oposed
al
g
o
r
ith
m
s
an
d
A-BF
wi
t
h
di
ffe
re
nt
val
u
e
s
o
f
m
an
d
N
Fi
gu
re
2.
C
o
m
m
uni
cat
i
on cos
t
s wi
t
h
di
f
f
ere
n
t
val
u
es
o
f
N
a
n
d
di
ffe
rent
va
l
u
es
of
m
5.
CO
NCL
USI
O
N
In
t
h
i
s
pap
e
r a
n
ovel
f
r
am
ewo
r
k
i
s
pr
o
p
o
s
ed fo
r ran
k
i
n
g di
st
ri
b
u
t
e
d
unce
r
t
a
i
n
dat
a
base.
The
fram
e
wo
rk
consists o
f
th
ree
main
layers (
Qu
ery
;
Ranking
and
Mon
ito
ring
), each one
has its own algorithm
s
.
The m
a
i
n
cont
ri
b
u
t
i
on
of t
h
i
s
pape
r i
s
at
bot
h
R
anki
n
g
a
nd
Mon
ito
ring
layers. Two
n
o
v
e
l algo
rithm
s
fo
r
ran
k
i
n
g di
st
ri
b
u
t
e
d u
n
ce
rt
ai
n dat
a
base are
pr
op
ose
d
at
Ranking
layer. Exp
ected
rank
is th
e rank
ing
fun
c
tion
u
s
ed
in
bo
th
alg
o
rith
m
s
. Th
e first alg
o
rith
m
(TDT
k
) u
tilizes o
n
l
y on
e co
mm
u
n
i
catio
n
roun
d
with
m
x
k
t
upl
e
s
sent
t
o
que
ry
s
i
t
e
. The
seco
n
d
on
e (C
DT
k
) u
tilizes
two
commu
n
i
catio
n
ro
und
s with
on
l
y
(
k + m
) tupl
es, at
m
o
st, in
th
e
first ro
und
. An exp
e
rim
e
n
t
al stud
y is m
a
de to test t
h
e
efficiency
a
nd effectivene
s
s
of t
h
e
pr
o
pose
d
al
go
r
i
t
h
m
s
agai
nst
A-BF
al
go
ri
t
h
m
[26]
.
Fr
om
obs
er
vi
n
g
t
h
e
expe
ri
m
e
nt
s resul
t
s
, i
t
i
s
cl
ear t
h
at
TDT
k
ov
ercomes CDTk
i
n
so
m
e
situ
atio
n
s
wh
ile CDT
k
ov
er
co
m
e
s TD
T
k
i
n
ot
her
s
i
t
u
at
i
ons,
an
d bot
h o
f
t
h
em
overc
om
e t
h
e
ol
d
o
n
e.
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